{"id":808,"date":"2021-12-14T18:34:44","date_gmt":"2021-12-14T18:34:44","guid":{"rendered":"https:\/\/www.ethanepperly.com\/?p=808"},"modified":"2025-09-12T20:08:01","modified_gmt":"2025-09-12T20:08:01","slug":"the-elegant-geometry-of-the-generalized-eigenvalue-perturbation-theory","status":"publish","type":"post","link":"https:\/\/www.ethanepperly.com\/index.php\/2021\/12\/14\/the-elegant-geometry-of-the-generalized-eigenvalue-perturbation-theory\/","title":{"rendered":"The Elegant Geometry of Generalized Eigenvalue Perturbation Theory"},"content":{"rendered":"\n<p><\/p>\n\n\n\n<p>Unfortunately, Mathias and Li&#8217;s original paper\u2014on which this blog post is based\u2014appears to have been taken offline. I am uploading a copy here for reference:<\/p>\n\n\n\n<div data-wp-interactive=\"core\/file\" class=\"wp-block-file\"><object data-wp-bind--hidden=\"!state.hasPdfPreview\" hidden class=\"wp-block-file__embed\" data=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/12\/Mathias-and-Li-2004-The-definite-generalized-eigenvalue-problem-A-new.pdf\" type=\"application\/pdf\" style=\"width:100%;height:600px\" aria-label=\"Embed of Mathias and Li - 2004 - The definite generalized eigenvalue problem A new.\"><\/object><a id=\"wp-block-file--media-8e615d6d-f79f-420a-9c63-d3ca9b197587\" href=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/12\/Mathias-and-Li-2004-The-definite-generalized-eigenvalue-problem-A-new.pdf\">Mathias and Li &#8211; 2004 &#8211; The definite generalized eigenvalue problem A new<\/a><a href=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/12\/Mathias-and-Li-2004-The-definite-generalized-eigenvalue-problem-A-new.pdf\" class=\"wp-block-file__button wp-element-button\" download aria-describedby=\"wp-block-file--media-8e615d6d-f79f-420a-9c63-d3ca9b197587\">Download<\/a><\/div>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<p>In this post, I want to discuss a beautiful and simple geometric picture of the perturbation theory of definite generalized eigenvalue problems. As a culmination, we&#8217;ll see a taste of the beautiful perturbation theory of Mathias and Li, which appears to be not widely known in virtue of only being explained in a technical report. Perturbation theory for the generalized eigenvalue problem is a bit of a niche subject, but I hope you will stick around for some elegant arguments. In addition to explaining the mathematics, I hope this post serves as an allegory for the importance of having the right way of thinking about a problem; often, the solution to a seemingly unsolvable problem becomes almost self-evident when one has the right perspective.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">What is a Generalized Eigenvalue Problem?<\/h2>\n\n\n\n<p>This post is about the definite generalized eigenvalue problem, so it&#8217;s probably worth spending a few words talking about what generalized eigenvalue problems are and why you would want to solve them. Slightly simplifying some technicalities, a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Eigendecomposition_of_a_matrix#Generalized_eigenvalue_problem\">generalized eigenvalue problem<\/a> consists of finding nonzero vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> and a (possibly complex) numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c252618ceeb5905d69dcb7fffd94b651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"\/> such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8296a9c3eeac7fd88a90099cb1df4add_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#92;&#44;&#32;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"85\" style=\"vertical-align: 0px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"1\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-1\">1<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-1\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"1\">In an unfortunate choice of naming, there is actually a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Eigendecomposition_of_a_matrix#Generalized_eigenspaces\">completely different sense<\/a> in which it makes sense to talk about generalized eigenvectors, in the context of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Jordan_normal_form\">Jordan normal form<\/a> for standard eigenvalue problems.<\/span> The vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> is called an <em>eigenvector<\/em> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c252618ceeb5905d69dcb7fffd94b651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"\/> its <em>eigenvalue<\/em>. For our purposes, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> will be real <a href=\"https:\/\/en.wikipedia.org\/wiki\/Symmetric_matrix\">symmetric<\/a> (or even complex <a href=\"https:\/\/en.wikipedia.org\/wiki\/Hermitian_matrix\">Hermitian<\/a>) matrices; one can also consider generalized eigenvalue problemss for nonsymmetric and <a href=\"https:\/\/epubs.siam.org\/doi\/abs\/10.1137\/S0895479803428795?casa_token=Dt2x4rG6Bx0AAAAA:tPXzYbddIAIcBxvdjP3nzM6918so8Y5NqkvU9M4iAnmDtxsTrBSf4IN1P8l__8DsW8PZ7DV7pFd4\">even non-square<\/a> matrices <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>, but the symmetric case covers many applications of practical interest. The generalized eigenvalue problem is so-named because it <em>generalizes<\/em> the standard eigenvalue problem <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-92a9de9abff53ba94627c34b930db14d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: 0px;\"\/>, which is a special case of the generalized eigenvalue problem with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ae83fed4da99b3d54a6110aaa2e67e8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#32;&#61;&#32;&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"47\" style=\"vertical-align: 0px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"2\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-2\">2<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-2\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"2\">One can also further generalize the generalized eigenvalue problem to <a href=\"https:\/\/link.springer.com\/chapter\/10.1007\/978-3-319-15260-8_12\">polynomial<\/a> and <a href=\"https:\/\/en.wikipedia.org\/wiki\/Nonlinear_eigenproblem\">nonlinear<\/a> eigenvalue problems.<\/span>\n\n\n\n<p>Why might we want so solve a generalized eigenvalue problem? The applications are numerous (e.g., in <a href=\"https:\/\/www.sciencedirect.com\/science\/article\/abs\/pii\/0010465574900113\">chemistry<\/a>, <a href=\"https:\/\/arxiv.org\/abs\/1909.08925\">quantum computation<\/a>, <a href=\"https:\/\/ieeexplore.ieee.org\/document\/1102559\">systems and control theory<\/a>, etc.). My interest in perturbation theory for generalized eigenvalue problems arose <a href=\"https:\/\/arxiv.org\/abs\/2110.07492\">in the analysis of a quantum algorithm<\/a> for eigenvalue problems in chemistry, and the theory discussed in this article played a big role in that analysis. To give an application which is more easy to communicate than the quantum computation application which motivated my own interest, let&#8217;s discuss an application in classical mechanics.<\/p>\n\n\n\n<p>The <a href=\"https:\/\/en.wikipedia.org\/wiki\/Lagrangian_mechanics\">Lagrangian formalism<\/a> is a way of reformulating Newton&#8217;s laws of motion in a general coordinate system.<sup class=\"modern-footnotes-footnote \" data-mfn=\"3\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-3\">3<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-3\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"3\">The benefits of the Lagrangian framework are far deeper than working in generalized coordinate systems, but this is beyond the scope of our discussion and mostly beyond the scope of what I am knowledgeable enough to write meaningfully about.<\/span> If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1d83a8e8f99aa0fb16ebf30cc43326a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\"\/> denotes a vector of generalized coordinates describing our system and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-64631d73733b7fab044745cebd70959d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#111;&#116;&#123;&#113;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"8\" style=\"vertical-align: -4px;\"\/> denotes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1d83a8e8f99aa0fb16ebf30cc43326a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\"\/>&#8216;s time derivative, then the time evolution of a system with Lagrangian functional <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-68d5a6e98e5d0cb98a64c29477801433_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#76;&#40;&#113;&#44;&#92;&#100;&#111;&#116;&#123;&#113;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"50\" style=\"vertical-align: -5px;\"\/> are given by the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Euler\u2013Lagrange_equation\">Euler\u2013Lagrange equations<\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-54e5c220a44e1dc6b3a2c3f4b8296acb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#102;&#114;&#97;&#99;&#123;&#100;&#125;&#123;&#100;&#116;&#125;&#32;&#92;&#110;&#97;&#98;&#108;&#97;&#95;&#123;&#92;&#100;&#111;&#116;&#123;&#113;&#125;&#125;&#32;&#76;&#32;&#61;&#32;&#92;&#110;&#97;&#98;&#108;&#97;&#95;&#113;&#32;&#76;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"106\" style=\"vertical-align: -6px;\"\/>. If we choose <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1d83a8e8f99aa0fb16ebf30cc43326a2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"8\" style=\"vertical-align: -4px;\"\/> to represent the deviation of our system from equilibrium,<sup class=\"modern-footnotes-footnote \" data-mfn=\"4\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-4\">4<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-4\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"4\">That is, in our generalized coordinate system, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f94d6c18886c6dc56c79468d1daa2293_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\"\/> is a (static) equilibrium configuration for which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a6b938cf66e407b68dce236335cbc5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#100;&#111;&#116;&#123;&#113;&#125;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\"\/> whenever <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f94d6c18886c6dc56c79468d1daa2293_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#113;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-45ce8324ac882d42cff049bb5209bb90_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#111;&#116;&#123;&#113;&#125;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"41\" style=\"vertical-align: -4px;\"\/>.<\/span> then our Lagrangian is well-approximated by it&#8217;s second order Taylor series: <\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 36px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bc13a39cfdd12a778a5fe763f1492e77_l3.png\" height=\"36\" width=\"252\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#76;&#40;&#113;&#44;&#92;&#100;&#111;&#116;&#123;&#113;&#125;&#41;&#32;&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#76;&#95;&#48;&#32;&#43;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#32;&#113;&#94;&#92;&#116;&#111;&#112;&#32;&#65;&#32;&#113;&#32;&#45;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#125;&#123;&#50;&#125;&#32;&#92;&#100;&#111;&#116;&#123;&#113;&#125;&#94;&#92;&#116;&#111;&#112;&#32;&#66;&#92;&#100;&#111;&#116;&#123;&#113;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>By the Euler\u2013Lagrange equations, the equations of motion for small deviations from this equillibrium point are described by <\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 17px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-6c435de1cef45ac56748db3585248148_l3.png\" height=\"17\" width=\"86\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#66;&#92;&#100;&#100;&#111;&#116;&#123;&#113;&#125;&#32;&#61;&#32;&#45;&#65;&#113;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>A <a href=\"https:\/\/tutorial.math.lamar.edu\/classes\/de\/FundamentalSetsofSolutions.aspx\">fundamental set of solutions<\/a> of this system of differential equations is given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-714d7e6ca21edd4e6def52b3285eeec9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#101;&#125;&#94;&#123;&#92;&#112;&#109;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#125;&#32;&#92;&#44;&#32;&#116;&#125;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: 0px;\"\/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c252618ceeb5905d69dcb7fffd94b651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> are the generalized eigenvalues and eigenvectors of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"5\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-5\">5<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-5\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"5\">That is, all solutions to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b3b2c6f8aaf38b7de2041758751cb1ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#92;&#100;&#100;&#111;&#116;&#123;&#113;&#125;&#32;&#61;&#32;&#45;&#65;&#113;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"82\" style=\"vertical-align: -4px;\"\/> can be written (uniquely) as linear combinations of solutions of the form <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-714d7e6ca21edd4e6def52b3285eeec9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#101;&#125;&#94;&#123;&#92;&#112;&#109;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#45;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#125;&#32;&#92;&#44;&#32;&#116;&#125;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"67\" style=\"vertical-align: 0px;\"\/>.<\/span> In particular, if all the generalized eigenvalues are positive, then the equillibrium is stable and the square roots of the eigenvalues represent the modes of vibration. In the most simple mechanical systems, such as masses in one dimension connected by springs with the natural coordinate system, the matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is diagonal with diagonal entries equal to the different masses. In even slightly more complicated &#8220;freshman physics&#8221; problems, it is quite easy to find examples where, in the natural coordinate system, the matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is nondiagonal.<sup class=\"modern-footnotes-footnote \" data-mfn=\"6\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-6\">6<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-6\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"6\">Almost always, the matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is <a href=\"https:\/\/en.wikipedia.org\/wiki\/Definite_matrix\">positive definite<\/a>.<\/span> As this example shows, generalized eigenvalue problems aren&#8217;t crazy weird things since they emerge as natural descriptions of simple mechanical systems like coupled pendulums.<\/p>\n\n\n\n<p>One reason generalized eigenvalue problems aren&#8217;t more well-known is that one can easily reduce a generalized eigenvalue problem into a standard one. If the matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is invertible, then the generalized eigenvalues of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> are just the eigenvalues of the matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-cf3ca0fb096a8f12548ba36c6f275d84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#94;&#123;&#45;&#49;&#125;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"46\" style=\"vertical-align: 0px;\"\/>. For several reasons, this is a less-than-appealing way of reducing a generalized eigenvalue problem to a standard eigenvalue problem. A better way, appropriate when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are both symmetric and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is <a href=\"https:\/\/en.wikipedia.org\/wiki\/Definite_matrix\">positive definite<\/a>, is to reduce the generalized eigenvalue problem for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> to the symmetrically reduced matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3657b2ec39dad87932a9ba5a2a58416a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;&#65;&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: 0px;\"\/>, which also possesses the same eigenvalues as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>. In particular, the matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3657b2ec39dad87932a9ba5a2a58416a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;&#65;&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: 0px;\"\/> remains symmetric, which shows that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> has real eigenvalues by the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Spectral_theorem\">spectral theorem<\/a>. In the mechanical context, one can think of this reformulation as a change of coordinate system in which the &#8220;mass matrix&#8221; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> becomes the identity matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06a46a64abb2fc8a9d51631fef84fe19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#73;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/>.<\/p>\n\n\n\n<p>There are several good reasons to <em>not<\/em> simply reduce a generalized eigenvalue problem to a standard one, and perturbation theory gives a particular good reason. In order for us to change coordinates to change the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> matrix into an identity matrix, we must first <em>know<\/em> the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> matrix. If I presented you with an elaborate mechanical system which you wanted to study, you would need to perform measurements to determine the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> matrices. But all measurements are imperfect and the entries of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are inevitably corrupted by measurement errors. In the presence of these measurement errors, we must give up on computing the normal modes of vibration perfectly; we must content ourselves with computing the normal modes of vibration plus-or-minus some error term we hope we can be assured is small if our measurement errors are small. In this setting, reducing the problem to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3657b2ec39dad87932a9ba5a2a58416a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;&#65;&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: 0px;\"\/> seems less appealing, as I have to understand how the measurement errors in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are amplified in computing the triple product <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3657b2ec39dad87932a9ba5a2a58416a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;&#65;&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: 0px;\"\/>. This also suggests that computing <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3657b2ec39dad87932a9ba5a2a58416a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;&#65;&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: 0px;\"\/> may be a poor algorithmic strategy in practice: if the matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is <a href=\"https:\/\/en.wikipedia.org\/wiki\/Condition_number\">ill-conditioned<\/a>, there might be a great deal of error amplification in the computed product <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3657b2ec39dad87932a9ba5a2a58416a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;&#65;&#66;&#94;&#123;&#45;&#49;&#47;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"106\" style=\"vertical-align: 0px;\"\/>. One might hope that one might be able to devise algorithms with better <a href=\"https:\/\/en.wikipedia.org\/wiki\/Numerical_stability#Stability_in_numerical_linear_algebra\">numerical stability<\/a> properties if we don&#8217;t reduce the matrix pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> to a single matrix. This is not to say that reducing a generalized eigenvalue problem to a standard one isn&#8217;t a useful tool\u2014it definitely is. However, it is not something one should do reflexively. Sometimes, a generalized eigenvalue problem is best left as is and analyzed in its native form.<\/p>\n\n\n\n<p>The rest of this post will focus on the question <em>if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are real symmetric matrices (satisfying a definiteness condition, to be elaborated upon below), how do the eigenvalues of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9d910f405edea91c71febb30e1fa301a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#43;&#69;&#44;&#66;&#43;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"119\" style=\"vertical-align: -5px;\"\/> compare to those of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1653b8134b59e8fb4e789ac004d93957_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-7bf5d1207baa8be58658ce9d3cf12043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are small real symmetric perturbations?<\/em> In fact, it shall be no additional toil to handle the complex Hermitian case as well while we&#8217;re at it, so we shall do so. (Recall that a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Hermitian_matrix\">Hermitian matrix<\/a> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> satisfies <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a5083de17e26339c493d568789a727eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#94;&#42;&#32;&#61;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"58\" style=\"vertical-align: 0px;\"\/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f51ea49c8acd139ddaa5de2ac0e9e3f7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#99;&#100;&#111;&#116;&#41;&#94;&#42;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"24\" style=\"vertical-align: -5px;\"\/> is the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Conjugate_transpose\">conjugate transpose<\/a>. Since the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Complex_conjugate\">complex conjugate<\/a> does not change a real number, a real Hermitian matrix is necesarily symmetric <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4d8f53e4b1833311083d5133612a1330_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#94;&#42;&#32;&#61;&#32;&#65;&#94;&#92;&#116;&#111;&#112;&#32;&#61;&#32;&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"106\" style=\"vertical-align: 0px;\"\/>.) For the remainder of this post, let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1653b8134b59e8fb4e789ac004d93957_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-7bf5d1207baa8be58658ce9d3cf12043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> be Hermitian matrices of the same size. Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-440609a0ee14ff5d42d4ef0d7c075363_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#32;&#58;&#61;&#32;&#65;&#43;&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"91\" style=\"vertical-align: -2px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5ea5f948fff32ac4f49ea1b43a9ed09a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#32;&#58;&#61;&#32;&#66;&#32;&#43;&#32;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -2px;\"\/> denote the perturbations.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Symmetric Treatment<\/h2>\n\n\n\n<p>As I mentioned at the top of this post, our mission will really be to find the right way of thinking about perturbation theory for the generalized eigenvalue problem, after which the theory will follow much more directly than if we were to take a frontal assault on the problem. As we go, we shall collect nuggets of insight, each of which I hope will follow quite naturally from the last. When we find such an insight, we shall display it on its own line.<\/p>\n\n\n\n<p>The first insight is that we can think of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> interchangeably. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c252618ceeb5905d69dcb7fffd94b651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"\/> is a nonzero eigenvalue of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>, satisfying <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-292e32d75703278ae11603af79b05ea0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#92;&#44;&#32;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"85\" style=\"vertical-align: 0px;\"\/>, then <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5fab458cd77505b8b353ad323c46cb97_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#120;&#32;&#61;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#94;&#123;&#45;&#49;&#125;&#32;&#92;&#44;&#32;&#65;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"103\" style=\"vertical-align: 0px;\"\/>. That is, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-559bfbd310bb03498b91dc6627eaeae9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"27\" style=\"vertical-align: 0px;\"\/> is an eigenvalue of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8db53ead031b4c1884c377b585c14df1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#66;&#44;&#65;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>. Lack of symmetry is an ugly feature in a mathematical theory, so we seek to smooth it out. After thinking a little bit, notice that we can phrase the generalized eigenvalue condition symmetrically as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3691e33936c6dfd63fd5d2b804913491_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;&#92;&#44;&#32;&#65;&#120;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#44;&#32;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\"\/> with the associated eigenvalue being given by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c38d646489e8ff25efe6205fcf17aac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#47;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"\/>. This observation may seem trivial at first, but let us collect it for good measure.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>Treat <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> symmetrically by writing the eigenvalue as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c38d646489e8ff25efe6205fcf17aac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#47;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"\/> with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b6430230398f101d53bf8dc6e2b0e096_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#92;&#44;&#32;&#65;&#120;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#92;&#44;&#32;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"100\" style=\"vertical-align: -4px;\"\/>.<\/strong><\/p>\n\n\n\n<p>Before proceeding, let&#8217;s ask a question that, in our new framing, becomes quite natural: what happens when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-33bdbf60fa5de98f07555b3beb00638d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\"\/>? The case <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-33bdbf60fa5de98f07555b3beb00638d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\"\/> is problematic because it leads to a division by zero in the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c38d646489e8ff25efe6205fcf17aac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#47;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"\/>. However, if we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-dba78e487eaf37124d6889a588f01086_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"44\" style=\"vertical-align: -4px;\"\/>, this expression still makes sense: we&#8217;ve found a vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> for which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4ba008b34881b34984235bdc65aecca3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"56\" style=\"vertical-align: -4px;\"\/> but <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5c201b986374c81b1e3c235e08a145d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"57\" style=\"vertical-align: 0px;\"\/>. It makes sense to consider <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> still an eigenvector of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> with eigenvalue <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e2b53d46b651204b8d52170f2e062612_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#47;&#32;&#48;&#32;&#61;&#32;&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"\/>! Dividing by zero should justifiably make one squeemish, but it really is quite natural in the case to treat <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> as a genuine eigenvector with eigenvalue <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9f9e97344403ddcf19e9a85e05b98b52_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#105;&#110;&#102;&#116;&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"17\" style=\"vertical-align: 0px;\"\/>.<\/p>\n\n\n\n<p>Things get even worse if we find a vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> for which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4a20659fd7e93f9a1a82007f68d50bee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#66;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"104\" style=\"vertical-align: 0px;\"\/>. Then, any <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> can reasonably considered an eigenvalue of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-67acc04b00f8dc2d90969b62fd011df2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#44;&#32;&#66;&#120;&#32;&#61;&#32;&#48;&#32;&#61;&#32;&#92;&#98;&#101;&#116;&#97;&#92;&#44;&#32;&#65;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"132\" style=\"vertical-align: -4px;\"\/>. In such a case, <em>all<\/em> complex numbers are simultaneously eigenvalues of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>, in which case we call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> singular.<sup class=\"modern-footnotes-footnote \" data-mfn=\"7\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-7\">7<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-7\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"7\">More precisely, a pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is singular if the determinant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3277f0c4cbccd93285dfdaa6b7c3733f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#101;&#116;&#40;&#116;&#66;&#32;&#45;&#32;&#65;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"93\" style=\"vertical-align: -5px;\"\/> is identically zero for all <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d41df395cde0f29e3fd33e29ba4f8304_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: -1px;\"\/>.<\/span> For the generalized eigenvalue problem to make sense for a pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>, it is natural to require that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> not be singular. In fact, we shall assume an even stronger &#8220;definiteness&#8221; condition which ensures that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> has only real (or infinite) eigenvalues. Let us return to this question of definiteness in a moment and for now assume that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is not singular and possesses real eigenvalues.<\/p>\n\n\n\n<p>With this small aside taken care of, let us return to the main thread. By modeling eigenvalues as pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/>, we&#8217;ve traded one ugliness for another. While reformulating the eigenvalue as a pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> treats <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> symmetrically, it also adds an additional indeterminacy, scale. For instance, if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> is an eigenvalue of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>, then so is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-df58d297963dfee35ac8393ae15e5236_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#49;&#48;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#49;&#48;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"\/>. Thus, it&#8217;s better not to think of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> so much as a pair of numbers together with all of its possible scalings.<sup class=\"modern-footnotes-footnote \" data-mfn=\"8\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-8\">8<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-8\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"8\"><a href=\"https:\/\/en.wikipedia.org\/wiki\/Projective_space\">Projective space<\/a> provides a natural framework for studying such vectors up to scale indeterminacy.<\/span> For reasons that shall hopefully become more clear as we go forward, it will be helpful to only consider all the possible <strong>positive<\/strong> scalings of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/>\u2014e.g., all <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-60f7cace75f077bb9c80202a2f1cca8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#116;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#116;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -5px;\"\/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9357d7f074d343e8f5599e26b08d9a2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#32;&#62;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -2px;\"\/>. Geometrically, the set of all positive scalings of a point in two-dimensional space is precisely just a ray emanating from the origin.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>Represent eigenvalue pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> as rays emanating from the origin to account for scale ambiguity.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"228\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta-3-300x228.png\" alt=\"\" class=\"wp-image-897\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta-3-300x228.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta-3-1024x778.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta-3-768x584.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta-3-1536x1167.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta-3-2048x1556.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure><\/div>\n\n\n<p>Now comes a standard eigenvalue trick. It&#8217;s something you would never think to do originally, but once you see it once or twice you learn to try it as a matter of habit. The trick: multiply the eigenvalue-eigenvector relation by the (conjugate) transpose of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>:<sup class=\"modern-footnotes-footnote \" data-mfn=\"9\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-9\">9<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-9\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"9\">For another example of the trick, try applying it to the standard eigenvalue problem <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-92a9de9abff53ba94627c34b930db14d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: 0px;\"\/>. Multiplying by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-66f1dc519a73b0b6a120e278ce8be635_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> and rearranging gives <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c2a60de563852f6f08f32e00bf98086d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#65;&#120;&#47;&#120;&#94;&#42;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -5px;\"\/>\u2014the eigenvalue <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c252618ceeb5905d69dcb7fffd94b651_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"10\" style=\"vertical-align: 0px;\"\/> is equal to the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-14746f91e3728358a3cc70c634811b87_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#65;&#120;&#47;&#120;&#94;&#42;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"78\" style=\"vertical-align: -5px;\"\/>, which is so important it is given a name: the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Rayleigh_quotient\">Rayleigh quotient<\/a>. In fact, the largest and smallest eigenvalues of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> can be found by maximizing and minimizing the Rayleigh quotient.<\/span>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 41px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f989e44000d5e26273bf9e4214fb2a3d_l3.png\" height=\"41\" width=\"421\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#98;&#101;&#116;&#97;&#32;&#92;&#44;&#32;&#65;&#120;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#44;&#32;&#66;&#120;&#32;&#92;&#105;&#109;&#112;&#108;&#105;&#101;&#115;&#32;&#92;&#98;&#101;&#116;&#97;&#32;&#92;&#44;&#32;&#120;&#94;&#42;&#65;&#120;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#92;&#44;&#32;&#120;&#94;&#42;&#66;&#120;&#32;&#92;&#105;&#109;&#112;&#108;&#105;&#101;&#115;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#125;&#123;&#92;&#98;&#101;&#116;&#97;&#125;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#94;&#42;&#65;&#120;&#125;&#123;&#120;&#94;&#42;&#66;&#120;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>The above equation is highly suggestive: since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4e8e0f9c0be9d91a4d8967ea0d3374c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ed1d6be545a04e298950d10962399911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"11\" style=\"vertical-align: -4px;\"\/> are only determined up to a scaling factor, it shows we can take <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-cb553ff03749c0632fcd8aad44fd9b27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#65;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"76\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5a47139d7f62d0766acdc5195694daaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\"\/>. And by different scalings of the eigenvector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, we can scale <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ad07f8657fa833e06d1695cebf00c65a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#65;&#120;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"76\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-55b1d4498a64d9404a3664cdb58972cf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#66;&#120;&#32;&#61;&#32;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\"\/> by any positive factor we want. (This retroactively shows why it makes sense to only consider positive scalings of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4e8e0f9c0be9d91a4d8967ea0d3374c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ed1d6be545a04e298950d10962399911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"11\" style=\"vertical-align: -4px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"10\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-10\">10<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-10\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"10\">To make this point more carefully, we shall make great use of the identification between pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> and the pair of quadratic forms <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ee430eac658eda1210e11e7532297e35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#120;&#94;&#42;&#65;&#120;&#44;&#120;&#94;&#42;&#66;&#120;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"\/>. Thus, even though <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d2531bc57de51d50b5b8f55ca2e04285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#45;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#45;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"\/> lead to equivalent eigenvalues since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a60c9d04b4bad237d24a71ee0dfc4e1b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#47;&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#40;&#45;&#92;&#97;&#108;&#112;&#104;&#97;&#41;&#47;&#40;&#45;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"140\" style=\"vertical-align: -5px;\"\/>, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d2531bc57de51d50b5b8f55ca2e04285_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#45;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#45;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"70\" style=\"vertical-align: -5px;\"\/> don&#8217;t necessarily both arise from a pair of quadratic forms: if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-24037524aafc7df0a18f697f988583ab_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;&#32;&#61;&#32;&#40;&#120;&#94;&#42;&#32;&#65;&#120;&#44;&#120;&#94;&#42;&#32;&#66;&#120;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"171\" style=\"vertical-align: -5px;\"\/>, this does not mean there exists <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-7506eeeff09aad3bcf6b7259302df451_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#121;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: -4px;\"\/> such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-92ba018d59244f08b04a2658e8c05bcc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#121;&#94;&#42;&#32;&#65;&#121;&#44;&#121;&#94;&#42;&#32;&#66;&#121;&#41;&#32;&#61;&#32;&#40;&#45;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#45;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"195\" style=\"vertical-align: -5px;\"\/>. Therefore, we only consider <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-60f7cace75f077bb9c80202a2f1cca8f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#116;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#116;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"55\" style=\"vertical-align: -5px;\"\/> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9357d7f074d343e8f5599e26b08d9a2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#116;&#32;&#62;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"39\" style=\"vertical-align: -2px;\"\/>.<\/span>) The expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e277aac7d217920cd98398c3e0926fb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#65;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"41\" style=\"vertical-align: 0px;\"\/> is so important that we give it a name: the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Quadratic_form#Real_quadratic_forms\">quadratic form<\/a> (associated with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and evaluated at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>).<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>The eigenvalue pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> can be taken equal to the pair of quadratic forms <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ee430eac658eda1210e11e7532297e35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#120;&#94;&#42;&#65;&#120;&#44;&#120;&#94;&#42;&#66;&#120;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"103\" style=\"vertical-align: -5px;\"\/>.<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Complexifying Things<\/h2>\n\n\n\n<p>Now comes another standard mathematical trick: <em>represent points in two-dimensional space by complex numbers<\/em>. In particular, we identify the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> with the complex number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0471a28c0fc54a660cc5346fa4efd92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"11\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-11\">11<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-11\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"11\">Recall that we are assuming that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c95803c43dd6dc1cc1dcbef260c3a517_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#47;&#32;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"31\" style=\"vertical-align: -5px;\"\/> is real, so we can pick a scaling in which both <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4e8e0f9c0be9d91a4d8967ea0d3374c7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ed1d6be545a04e298950d10962399911_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"11\" style=\"vertical-align: -4px;\"\/> are real numbers. Assume we have done this.<\/span> Similar to the previous trick, it&#8217;s not fully clear why this will pay off, but let&#8217;s note it as an insight.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>Identify the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> with the complex number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0471a28c0fc54a660cc5346fa4efd92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/>.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"222\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta_complex-1-300x222.png\" alt=\"\" class=\"wp-image-894\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta_complex-1-300x222.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta_complex-1-1024x758.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta_complex-1-768x569.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta_complex-1-1536x1138.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/alpha_beta_complex-1-2048x1517.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure><\/div>\n\n\n<p>Now, we combine all the previous observations. The eigenvalue <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4fa36e2772ec71a1cb8bd45d75a63398_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#65;&#120;&#32;&#47;&#32;&#120;&#94;&#42;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"126\" style=\"vertical-align: -5px;\"\/> is best thought of as a pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> which, up to scale, can be taken to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-cb553ff03749c0632fcd8aad44fd9b27_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#65;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"76\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5a47139d7f62d0766acdc5195694daaf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"77\" style=\"vertical-align: -4px;\"\/>. But then we represent <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> as the complex number<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 19px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-72a25cf83c0bc70b56246230deef9eb6_l3.png\" height=\"19\" width=\"307\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#65;&#120;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#120;&#94;&#42;&#66;&#120;&#32;&#61;&#32;&#120;&#94;&#42;&#40;&#65;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#32;&#120;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>Let&#8217;s stop for a moment and appreciate how far we&#8217;ve come. The generalized eigenvalue problem <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-292e32d75703278ae11603af79b05ea0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#92;&#44;&#32;&#66;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"85\" style=\"vertical-align: 0px;\"\/> is associated with the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1d7d9ada0bc5820520ed8dc862c51f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/>.If we just went straight from one to the other, this reduction would appear like some crazy stroke of inspiration: why would I ever think to write down <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1d7d9ada0bc5820520ed8dc862c51f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/>? However, just following our nose lead by a desire to treat <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> symmetrically and applying a couple standard tricks, this expression appears naturally. The expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1d7d9ada0bc5820520ed8dc862c51f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> will be very useful to us because it is <em>linear<\/em> in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>, and thus for the perturbed problem <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-357fe78b9f13db750d370cb0ab42e5bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#44;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;&#32;&#61;&#32;&#40;&#65;&#43;&#69;&#44;&#66;&#43;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"192\" style=\"vertical-align: -5px;\"\/>, we have that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d3537253df3dd3073a017c720b621c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;&#120;&#32;&#61;&#32;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#120;&#32;&#43;&#32;&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#70;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"334\" style=\"vertical-align: -5px;\"\/>: consequently, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-79808efed3811d5adb73685ca5c582d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -5px;\"\/> is a small perturbation of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1d7d9ada0bc5820520ed8dc862c51f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/>. This observation will be very useful to us. <\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> is the eigenvector, then the complex number <\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-71c8c83e4f89fbddef51d84f497ecc70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/><strong> is <\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1d7d9ada0bc5820520ed8dc862c51f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/><strong>.<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Definiteness and the Crawford Number<\/h2>\n\n\n\n<p>With these insights in hand, we can now return to the point we left earlier about what it means for a generalized eigenvalue problem to be &#8220;definite&#8221;. We know that if there exists a vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> for which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4a20659fd7e93f9a1a82007f68d50bee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#66;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"104\" style=\"vertical-align: 0px;\"\/>, then the problem is singular. If we multiply by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-66f1dc519a73b0b6a120e278ce8be635_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/>, we see that this means that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d20fc47af274231b2c038c763e99c0da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#65;&#120;&#32;&#61;&#32;&#120;&#94;&#42;&#66;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"140\" style=\"vertical-align: 0px;\"\/> as well and thus <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-76a3cd51c86035eff687b49cdb4afc0d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"\/>. It is thus quite natural to assume the following definiteness condition:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>The pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is said to be <strong>definite<\/strong> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-87600d43aa97e51c1aae3f4bba2f894c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#32;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"\/> for all complex nonzero vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<\/blockquote>\n\n\n\n<p>A definite problem is guaranteed to be not singular, but the reverse is not necessarily true; one can easily find pairs <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> which are not definite and also not singular.<sup class=\"modern-footnotes-footnote \" data-mfn=\"12\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-12\">12<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-12\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"12\">For example, consider <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-88a60a51171cac2245ae8a8f56611fee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#32;&#61;&#32;&#66;&#32;&#61;&#32;&#92;&#111;&#112;&#101;&#114;&#97;&#116;&#111;&#114;&#110;&#97;&#109;&#101;&#123;&#100;&#105;&#97;&#103;&#125;&#40;&#49;&#44;&#45;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"160\" style=\"vertical-align: -5px;\"\/>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is not definite since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d20fc47af274231b2c038c763e99c0da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#65;&#120;&#32;&#61;&#32;&#120;&#94;&#42;&#66;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"140\" style=\"vertical-align: 0px;\"\/> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-194e9728f5eeb9e101e7b31d599dee21_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#32;&#61;&#32;&#40;&#49;&#44;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"72\" style=\"vertical-align: -5px;\"\/>. However, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is not singular; the only eigenvalue of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-27e3598fd2d4f0491067f1afaced92e9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"7\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-7ea0dc3e3ba70ac0ef7c822497749bc6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#100;&#101;&#116;&#40;&#116;&#66;&#45;&#65;&#41;&#32;&#61;&#32;&#45;&#40;&#116;&#45;&#49;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"181\" style=\"vertical-align: -5px;\"\/> is not identically zero..<\/span> (Note <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d20fc47af274231b2c038c763e99c0da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#65;&#120;&#32;&#61;&#32;&#120;&#94;&#42;&#66;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"140\" style=\"vertical-align: 0px;\"\/> does not imply <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4a20659fd7e93f9a1a82007f68d50bee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#120;&#32;&#61;&#32;&#66;&#120;&#32;&#61;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"104\" style=\"vertical-align: 0px;\"\/> unless <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are <em>both<\/em> positive (or negative) semidefinite.)<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>The &#8220;natural&#8221; symmetric condition for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> to be &#8220;definite&#8221; is for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4c6d140c9d055899e947acdf9929298f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#66;&#41;&#120;&#32;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"\/> for all vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>.<\/strong><\/p>\n\n\n\n<p>Since the expression <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> is just scaled by a positive factor by scaling the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>, it is sufficient to check the definiteness condition <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-87600d43aa97e51c1aae3f4bba2f894c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#32;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"129\" style=\"vertical-align: -5px;\"\/> for only complex <em>unit<\/em> vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>. This leads naturally to a quantitative characterization of the degree of definiteness of a pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>The <strong>Crawford number<\/strong><sup class=\"modern-footnotes-footnote \" data-mfn=\"13\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-13\">13<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-13\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"13\">The name <em>Crawford number<\/em> was coined by <a href=\"http:\/\/users.umiacs.umd.edu\/~stewart\/\">G. W. Stewart<\/a> in <a href=\"http:\/\/www.sciencedirect.com\/science\/article\/pii\/0024379579900946\">1979<\/a> in recognition of <a href=\"http:\/\/epubs.siam.org\/doi\/abs\/10.1137\/0713067\">Crawford&#8217;s pioneering work<\/a> on the perturbation theory of the definite generalized eigenvalue problem.<\/span> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> of a pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is the minimum value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1ce78711e7a3d3cfcc797ebbd1546920_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#124;&#32;&#61;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#40;&#120;&#94;&#42;&#65;&#120;&#41;&#94;&#50;&#32;&#43;&#32;&#40;&#120;&#94;&#42;&#66;&#120;&#41;&#94;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"294\" style=\"vertical-align: -6px;\"\/> over all complex unit vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>.<\/p>\n<\/blockquote>\n\n\n\n<p>The Crawford number naturally quantifies the degree of definiteness.<sup class=\"modern-footnotes-footnote \" data-mfn=\"14\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-14\">14<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-14\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"14\">In fact, <a href=\"https:\/\/epubs.siam.org\/doi\/abs\/10.1137\/040616346\">it has been shown<\/a> that the Crawford number is, in a sense, the distance from a definite matrix pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> to a pair which is not simultaneously diagonalizable by congruence.<\/span> A problem which has a large Crawford number (relative to a perturbation) will remain definite after perturbation, whereas the pair may become indefinite if the size of the perturbation exceeds the Crawford number. Geometrically, the Crawford number has the following interpretation: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> must lie on or outside the circle of radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-54589d9b5610bf48dcf5a1b1f24a67b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> for all (complex) unit vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>The &#8220;degree of definiteness&#8221; can be quantified by the Crawford number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-094268968f074367f504b644c09c0296_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;&#32;&#58;&#61;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#32;&#120;&#94;&#42;&#40;&#65;&#43;&#105;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"255\" style=\"vertical-align: -8px;\"\/>.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-medium is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"300\" height=\"238\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/crawford-1-300x238.png\" alt=\"\" class=\"wp-image-893\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/crawford-1-300x238.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/crawford-1-1024x813.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/crawford-1-768x610.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/crawford-1-1536x1220.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/crawford-1-2048x1626.png 2048w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/figure><\/div>\n\n\n<p>Now comes another step in our journey which is a bit more challenging. For a matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-52134c3741ef3371f17ceb962d0792f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> (in our case <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8f0b1062a3c941044226e23b2307a03a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;&#32;&#61;&#32;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"92\" style=\"vertical-align: -2px;\"\/>), the set of complex numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e629538fe34ef28eff91f709c28d4fbb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#67;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"42\" style=\"vertical-align: 0px;\"\/> for all unit vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> has been the subject of considerable study. In fact, this set has a name<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>The <strong>field of values<\/strong> of a matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-52134c3741ef3371f17ceb962d0792f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is the set <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-808adb25fdeba07c406a51eb1c85fc1e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#40;&#67;&#41;&#32;&#58;&#61;&#32;&#92;&#123;&#32;&#120;&#94;&#42;&#67;&#120;&#32;&#58;&#32;&#120;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#67;&#125;&#94;&#110;&#44;&#32;&#92;&#58;&#32;&#92;&#124;&#120;&#92;&#124;&#32;&#61;&#32;&#49;&#92;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"275\" style=\"vertical-align: -5px;\"\/>.<\/p>\n<\/blockquote>\n\n\n\n<p>In particular, the Crawford number is just the absolute value of the closest complex number in the field of values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-22f690df49941e51c062cb7414b6253d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#40;&#65;&#43;&#105;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"88\" style=\"vertical-align: -5px;\"\/> to zero.<\/p>\n\n\n\n<p>It is a very cool and highly nontrivial fact (called the <a href=\"https:\/\/www.cambridge.org\/core\/journals\/canadian-mathematical-bulletin\/article\/toeplitzhausdorff-theorem-explained\/BA251EBB1E1DE08DBD3D84964F65938B\">Toeplitz\u2013Hausdorff Theorem<\/a>) that the field of values is always a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Convex_set\">convex set<\/a>, with every two points in the field of values containing the line segment connecting them. Thus, as a consequence, the field of values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-891cec293aa95cacafe85ffe91032b8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"\/> for a definite matrix pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> is always on &#8220;one side&#8221; of the complex plane (in the sense that there exists a line through zero which <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-891cec293aa95cacafe85ffe91032b8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"\/> lies strictly on one side of<sup class=\"modern-footnotes-footnote \" data-mfn=\"15\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-15\">15<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-15\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"15\">This is a consequence of the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Hyperplane_separation_theorem\">hyperplane separation theorem<\/a> together with the fact that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0bab004479a0111974cedf5d3251f77d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#110;&#111;&#116;&#105;&#110;&#32;&#87;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"117\" style=\"vertical-align: -5px;\"\/>.<\/span>).<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>The numbers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-79164a2a979436867c0cdffdb4e235d3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#105;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"97\" style=\"vertical-align: -5px;\"\/> for unit vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> lie on one half of the complex plane.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"627\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/field_of_values-1024x627.png\" alt=\"\" class=\"wp-image-892\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/field_of_values-1024x627.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/field_of_values-300x184.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/field_of_values-768x470.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/field_of_values-1536x940.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/field_of_values-2048x1254.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">The field of values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-891cec293aa95cacafe85ffe91032b8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"\/> lies outside the circle of radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-54589d9b5610bf48dcf5a1b1f24a67b6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> and thus on one side of the complex plane.<\/figcaption><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\">From Eigenvalues to Eigenangles<\/h2>\n\n\n\n<p>All of this geometry is lovely, but we need some way of relating it to the eigenvalues. As we observed a while ago, each eigenvalue is best thought of as a <strong>ray<\/strong> emanating from the origin, owing to the fact that the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> can be scaled by an arbitrary positive factor. A ray is naturally associated with an <em>angle<\/em>, so it is natural to characterize an eigenvalue pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> by the angle describing its arc.<\/p>\n\n\n\n<p>But the angle of a ray is only defined up additions by full rotations (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f9f114333dea6546058b6b821dbb3910_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#112;&#105;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"20\" style=\"vertical-align: 0px;\"\/> radians). As such, to associate each ray a unique angle we need to pin down this indeterminacy in some fixed way. Moreover, this indeterminacy should play nice with the field of values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-891cec293aa95cacafe85ffe91032b8c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"86\" style=\"vertical-align: -5px;\"\/> <em>and<\/em> the field of values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8df1422122b0325a2db67586d00f2c22_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#87;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"86\" style=\"vertical-align: -5px;\"\/> of the perturbation. But in the last section, we say that each of these field of angles lies (strictly) on one half of the complex plane. Thus, we can find a ray <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d08fac616919760e7538df715d3ca0e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> which does not intersect either field of values!<\/p>\n\n\n\n<p>One possible choice is to measure the angle from this ray. We shall make a slightly different choice which plays better when we treat <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> as a complex number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0471a28c0fc54a660cc5346fa4efd92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/>. Recall that a number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3eff9dcbcd5c19f0c719ef58060e0716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> is an <strong><a href=\"https:\/\/en.wikipedia.org\/wiki\/Argument_(complex_analysis)\">argument<\/a><\/strong> for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0471a28c0fc54a660cc5346fa4efd92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/> if <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-679486ec9d570cc09f2ec1d1d296ff5a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#114;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#101;&#125;&#94;&#123;&#105;&#92;&#116;&#104;&#101;&#116;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"101\" style=\"vertical-align: -4px;\"\/> for some real number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-378f532521e3b95e04ca61f4db1b3ba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#32;&#92;&#103;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"41\" style=\"vertical-align: -3px;\"\/>. The argument is multi-valued since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1ff07bcc081bfdba552070da5b1f396e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#43;&#32;&#50;&#92;&#112;&#105;&#32;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"61\" style=\"vertical-align: -2px;\"\/> is an argument for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0e05f8a5dedd123b1ba59cd40c9955a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/> as long as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3eff9dcbcd5c19f0c719ef58060e0716_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"9\" style=\"vertical-align: 0px;\"\/> is (for all integers <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75a5652acadcd645b180a972b75a9d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>). However, once we exclude our ray <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d08fac616919760e7538df715d3ca0e6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>, we can assign each complex number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0e05f8a5dedd123b1ba59cd40c9955a9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/> not on this ray a unique argument which depends continuously on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/>. Denote this &#8220;<a href=\"https:\/\/en.wikipedia.org\/wiki\/Branch_point#Branch_cuts\">branch<\/a>&#8221; of the argument by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-cbaf19b809e7268b72f25042f2e3d25d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#112;&#101;&#114;&#97;&#116;&#111;&#114;&#110;&#97;&#109;&#101;&#123;&#97;&#114;&#103;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"25\" style=\"vertical-align: -4px;\"\/>. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> represents an eigenvalue <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c38d646489e8ff25efe6205fcf17aac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#47;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"\/>, we call <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d88ab65f0fb855191702f565da6d8f45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#32;&#92;&#97;&#114;&#103;&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"119\" style=\"vertical-align: -5px;\"\/> an <strong>eigenangle<\/strong>.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>Represent an eigenvalue pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> by its associated eigenangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b750d4fb213695b4855240ecadaab02e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#32;&#92;&#97;&#114;&#103;&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#105;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"120\" style=\"vertical-align: -5px;\"\/>.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"553\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/eigenangle-3-1024x553.png\" alt=\"\" class=\"wp-image-903\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/eigenangle-3-1024x553.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/eigenangle-3-300x162.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/eigenangle-3-768x415.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/eigenangle-3-1536x829.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/eigenangle-3-2048x1106.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n<p>How are these eigenangles related to the eigenvalues? It&#8217;s now a trigonometry problem:<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 40px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-82ba1edcd8710cf25e03f1ec940551a9_l3.png\" height=\"40\" width=\"299\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#97;&#108;&#112;&#104;&#97;&#125;&#123;&#92;&#98;&#101;&#116;&#97;&#125;&#32;&#61;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#109;&#98;&#111;&#120;&#123;&#97;&#100;&#106;&#97;&#99;&#101;&#110;&#116;&#125;&#125;&#123;&#92;&#109;&#98;&#111;&#120;&#123;&#111;&#112;&#112;&#111;&#115;&#105;&#116;&#101;&#125;&#125;&#32;&#61;&#32;&#92;&#99;&#111;&#116;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#92;&#111;&#112;&#101;&#114;&#97;&#116;&#111;&#114;&#110;&#97;&#109;&#101;&#123;&#97;&#114;&#103;&#125;&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#105;&#92;&#98;&#101;&#116;&#97;&#41;&#41;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>The eigenvalues are the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Trigonometric_functions#cot\">cotangents<\/a> of the eigenangles!<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>The eigenvalue <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c38d646489e8ff25efe6205fcf17aac7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#32;&#61;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#47;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"65\" style=\"vertical-align: -5px;\"\/> is the cotangent of the eigenangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d88ab65f0fb855191702f565da6d8f45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#61;&#32;&#92;&#97;&#114;&#103;&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"119\" style=\"vertical-align: -5px;\"\/>.<\/strong><\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Variational Characterization<\/h2>\n\n\n\n<p>Now comes another difficulty spike in our line of reasoning, perhaps the largest in our whole deduction. To properly motivate things, let us first review some facts about the standard Hermitian\/symmetric eigenvalue problem. The big idea is that eigenvalues can be thought of as the solution to a certain optimization problem. The largest eigenvalue of a Hermitian\/symmetric matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> is given by the maximization problem<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 30px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-45dd37a25f32afaf3e581e8a371d899f_l3.png\" height=\"30\" width=\"174\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#123;&#92;&#114;&#109;&#32;&#109;&#97;&#120;&#125;&#40;&#65;&#41;&#32;&#61;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#124;&#120;&#92;&#124;&#32;&#61;&#32;&#49;&#125;&#32;&#120;&#94;&#42;&#65;&#120;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>The largest eigenvalue is the maximum of the quadratic form over unit vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>. What about the other eigenvalues? The answer is not obvious, but the famous <a href=\"https:\/\/en.wikipedia.org\/wiki\/Min-max_theorem#Min-max_theorem\">Courant\u2013Fischer Theorem<\/a> shows that the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>th largest eigenvalue <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ab4ad5c7f466d9063d4866d8a737624b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#106;&#40;&#65;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"43\" style=\"vertical-align: -6px;\"\/> can be written as the following minimax optimization problem<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 42px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-13a869b76f612be23e50bc9e0897ee1c_l3.png\" height=\"42\" width=\"248\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#106;&#40;&#65;&#41;&#32;&#61;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#100;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#61;&#32;&#110;&#45;&#106;&#43;&#49;&#125;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#32;&#61;&#32;&#49;&#125;&#125;&#32;&#120;&#94;&#42;&#65;&#120;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>The minimum is taken over all <em>subspaces<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ef4ca273d35c61336fca8006b215f82c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"\/> of dimension <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f67f89012cdf1bb42128176fdeb0d659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#45;&#106;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\"\/> whereas the maximum is taken over all unit vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> within the subspace <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ef4ca273d35c61336fca8006b215f82c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"\/>. Symmetrically, one can also formulate the eigenvalues as a max-min optimization problem<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 42px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9d7fb9357b6c4cd442d40084b1301210_l3.png\" height=\"42\" width=\"212\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#106;&#40;&#65;&#41;&#32;&#61;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#100;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#61;&#32;&#106;&#125;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#32;&#61;&#32;&#49;&#125;&#125;&#32;&#120;&#94;&#42;&#65;&#120;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>These variational\/minimax characterizations of the eigenvalues of a Hermitian\/symmetric matrix are essential to perturbation theory for Hermitian\/symmetric eigenvalue problems, so it is only natural to go looking for a variational characterization of the generalized eigenvalue problem. There is one natural way of doing this that works for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> positive definite: specifically, one can show that<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 53px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75f89c3046a9ebf3f8ddb29e50d7af45_l3.png\" height=\"53\" width=\"276\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#106;&#40;&#65;&#44;&#66;&#41;&#32;&#61;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#100;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#61;&#32;&#110;&#45;&#106;&#43;&#49;&#125;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#32;&#61;&#32;&#49;&#125;&#125;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#120;&#94;&#42;&#65;&#120;&#125;&#123;&#120;&#94;&#42;&#66;&#120;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>This characterization, while useful in its own right, is tricky to deal with because it is nonlinear in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>. It also treats <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> non-symmetrically, which should set off our alarm bells that there might be a better way. Indeed, the ingenious idea, due to G. W. Stewart in 1979, is to instead provide a variational characterization of the <em>eigenangles<\/em>! Specifically, Stewart was able to show<sup class=\"modern-footnotes-footnote \" data-mfn=\"16\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-16\">16<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-16\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"16\">Stewart&#8217;s original definition of the eigenangles differs from ours; we adopt the definition of Mathias and Li. The result amounts to the same thing.<\/span>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 93px;\"><span class=\"ql-right-eqno\"> (1) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a448bd6bb114bcc4e570431ef11dd72_l3.png\" height=\"93\" width=\"313\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#38;&#61;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#100;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#61;&#32;&#110;&#45;&#106;&#43;&#49;&#125;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#125;&#32;&#92;&#97;&#114;&#103;&#40;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#41;&#44;&#32;&#92;&#92; &#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#38;&#61;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#100;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#61;&#32;&#106;&#125;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#125;&#32;&#92;&#97;&#114;&#103;&#40;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#41;&#44; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>for the eigenangles <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4739d6722990ac5d4215734b678d6512_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#49;&#92;&#103;&#101;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#50;&#92;&#103;&#101;&#92;&#99;&#100;&#111;&#116;&#115;&#92;&#103;&#101;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"140\" style=\"vertical-align: -3px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"17\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-17\">17<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-17\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"17\">Note that since the cotangent is <em>decreasing<\/em> on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-37f5d6c600452a51b9fbfc23122f4a36_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#091;&#48;&#44;&#92;&#112;&#105;&#093;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"33\" style=\"vertical-align: -5px;\"\/>, this means that the eigenvalues <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e422c56d4d4d9829a77f820bb6afc2da_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#49;&#32;&#61;&#32;&#92;&#99;&#111;&#116;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#49;&#32;&#92;&#108;&#101;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#50;&#32;&#61;&#32;&#92;&#99;&#111;&#116;&#32;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#50;&#32;&#92;&#108;&#101;&#32;&#92;&#99;&#100;&#111;&#116;&#115;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"238\" style=\"vertical-align: -3px;\"\/> are now in <em>increasing<\/em> order, in contrast to our convention from earlier in this section.<\/span> This shows, in particular, that the field of values is subtended by the smallest and largest eigenangles.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>The eigenangles satisfy a minimax variational characterization.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"683\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/variational-1024x683.png\" alt=\"\" class=\"wp-image-904\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/variational-1024x683.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/variational-300x200.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/variational-768x512.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/variational-1536x1024.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/variational-2048x1366.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n<h2 class=\"wp-block-heading\">How Big is the Perturbation?<\/h2>\n\n\n\n<p>We&#8217;re tantalizingly close to our objective. The final piece in our jigsaw puzzle before we&#8217;re able to start proving perturbation theorems is to quantify the size of the perturbing matrices <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1653b8134b59e8fb4e789ac004d93957_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-7bf5d1207baa8be58658ce9d3cf12043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>. Based on what we&#8217;ve done so far, we see that the eigenvalues are natural associated with the complex number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/>, so it is natural to characterize the size of the perturbing pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-653ca903a805276f0592e95ae97d308f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#69;&#44;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\"\/> by the distance between <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-dd8d37f0240332e72d408d0cece8ae0f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"96\" style=\"vertical-align: -5px;\"\/>. But the difference between these two quantities is just <\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 23px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c94fbeea27a6f4e3ae10e57de1aed2b5_l3.png\" height=\"23\" width=\"337\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#120;&#94;&#42;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;&#120;&#32;&#45;&#32;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#32;&#61;&#32;&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>We&#8217;re naturally led to the question: how big can <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fda32ca3ea08f20eea6e67bee075b56c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> be? If the vector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> has a large norm, then quite large, so let&#8217;s fix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> to be a unit vector. With this assumption in place, the maximum size of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fda32ca3ea08f20eea6e67bee075b56c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> is simple the distance of the <em>farthest<\/em> point in the field of values <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8bb67ee1f134a1524e3a124ceaeeeb5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#43;&#105;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"56\" style=\"vertical-align: -2px;\"\/> from zero. This quantity has a name:<\/p>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<p>The <strong>numerical radius<\/strong> of a matrix <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-53c2a72f422a8e6f02f2e3ac92c66d78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#71;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> (in our case <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e6d024fca992cb665f00fe058ef239f8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#61;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"73\" style=\"vertical-align: -2px;\"\/>) is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2e378dbebd1cd357504d7615feb843d9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#71;&#41;&#32;&#58;&#61;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#32;&#124;&#120;&#94;&#42;&#71;&#120;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"190\" style=\"vertical-align: -8px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"18\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-18\">18<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-18\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"18\">This maximum is taken over all <em>complex<\/em> unit vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>.<\/span>\n<\/blockquote>\n\n\n\n<p class=\"has-text-align-center\"><strong>The size of the perturbation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-653ca903a805276f0592e95ae97d308f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#69;&#44;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\"\/> is the numerical radius <\/strong><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-92fb1e32ba076fcd54dcfe982adbeed1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#61;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#32;&#124;&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"281\" style=\"vertical-align: -8px;\"\/><strong>.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"712\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/numerical_radius-1-1024x712.png\" alt=\"\" class=\"wp-image-908\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/numerical_radius-1-1024x712.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/numerical_radius-1-300x209.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/numerical_radius-1-768x534.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/numerical_radius-1-1536x1068.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/numerical_radius-1-2048x1424.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n<p>It is easy to upper-bound the numerical radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> by more familiar quantities. For instance, once can straightforwardly show the bound <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ef407c1cb7121439ba06449ee040b810_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#92;&#108;&#101;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#92;&#124;&#69;&#92;&#124;&#94;&#50;&#43;&#92;&#124;&#70;&#92;&#124;&#94;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"219\" style=\"vertical-align: -6px;\"\/>, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-6f7c7374d9450ba3b092597ca402f7b1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#124;&#92;&#99;&#100;&#111;&#116;&#92;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"27\" style=\"vertical-align: -5px;\"\/> is the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Matrix_norm#Spectral_norm\">spectral norm<\/a>. We prefer to state results using the numerical radius because of its elegance: it is, in some sense, the &#8220;correct&#8221; measure of the size of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-653ca903a805276f0592e95ae97d308f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#69;&#44;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\"\/> in the context of this theory.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Stewart&#8217;s Perturbation Theory<\/h2>\n\n\n\n<p>Now, after many words of prelude, we finally get to our first perturbation theorem. With the work we&#8217;ve done in place, the result is practically effortless.<\/p>\n\n\n\n<p>Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-6141e3bcc5f2053fcf9bad0cc6ac1980_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#49;&#92;&#103;&#101;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#50;&#92;&#103;&#101;&#32;&#92;&#99;&#100;&#111;&#116;&#115;&#92;&#103;&#101;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"140\" style=\"vertical-align: -3px;\"\/> denote the eigenangles of the perturbed pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fda90273726cf242b0edb2bf5db8a316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#44;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -5px;\"\/> and consider the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>th eigenangle. Let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-483085c7b6b0c5786a1f1f034eee5793_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#94;&#42;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\"\/> be the subspace of dimension <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f67f89012cdf1bb42128176fdeb0d659_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#45;&#106;&#43;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"70\" style=\"vertical-align: -4px;\"\/> achieving the minimum in the first equation of the variational principle (1) for the <em>original unperturbed pair<\/em> <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>. Then we have<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 46px;\"><span class=\"ql-right-eqno\"> (2) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a13a27463f577615e4129b59ed2ac75b_l3.png\" height=\"46\" width=\"601\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#61;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#100;&#105;&#109;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#61;&#32;&#110;&#45;&#106;&#43;&#49;&#125;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#125;&#32;&#92;&#97;&#114;&#103;&#40;&#120;&#94;&#42;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;&#120;&#41;&#32;&#92;&#108;&#101;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#94;&#42;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#125;&#32;&#92;&#97;&#114;&#103;&#40;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#32;&#43;&#32;&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;&#41;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>This is something of a standard trick when dealing with variational problems in matrix analysis: take the solution (in this case the minimizing subspace) for the original problem and plug it in for the perturbed problem. The solution may no longer be optimal, but it at least gives an upper (or lower) bound. The complex number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> must lie at least a distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> from zero and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-535a73024bc9ce80afc8c592cea154a6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;&#124;&#32;&#92;&#108;&#101;&#32;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"204\" style=\"vertical-align: -5px;\"\/>. We&#8217;re truly toast if the perturbation is large enough to perturb <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a17a92e32974279275a4862c4503a5a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#105;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"97\" style=\"vertical-align: -5px;\"\/> to be equal to zero, so we should assume that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8279147aba308945f984cfd5590ebc94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#60;&#32;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -5px;\"\/>.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>For our perturbation theory to work, we must assume <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8279147aba308945f984cfd5590ebc94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#60;&#32;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -5px;\"\/>.<\/strong><\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"826\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/definiteness_condition-1024x826.png\" alt=\"\" class=\"wp-image-910\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/definiteness_condition-1024x826.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/definiteness_condition-300x242.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/definiteness_condition-768x620.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/definiteness_condition-1536x1240.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/definiteness_condition-2048x1653.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> lies on or outside the circle centered at zero with radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/>. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e8fefe41fe5fe4721833eae109ab876f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#92;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"96\" style=\"vertical-align: -5px;\"\/> might lie anywhere in a circle centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> with radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/>, so one must have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8279147aba308945f984cfd5590ebc94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#60;&#32;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -5px;\"\/> to ensure the perturbed problem is nonsingular (equivalently <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ad852c327346ebf30979963e2a226a45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#92;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#92;&#110;&#101;&#32;&#48;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"129\" style=\"vertical-align: -5px;\"\/> for every <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>).<\/figcaption><\/figure><\/div>\n\n\n<p>Making the assumption that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8279147aba308945f984cfd5590ebc94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#60;&#32;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -5px;\"\/>, bounding the right-hand side of (2) requires finding the most-counterclockwise angle necessary to subtend a circle of radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> centered at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/>, which must lie a distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> from the origin. The worst-case scenario is when <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> is exactly a distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> from the origin, as is shown in the following diagram.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"729\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/stewart-1024x729.png\" alt=\"\" class=\"wp-image-913\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/stewart-1024x729.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/stewart-300x213.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/stewart-768x546.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/stewart-1536x1093.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/stewart-2048x1457.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">In the worst case, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> lies on the circle centered at zero with radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/>, which is subtended above by angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8ed6b13dc005290747ef02210c94a981_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#115;&#105;&#110;&#94;&#123;&#45;&#49;&#125;&#40;&#114;&#40;&#69;&#43;&#105;&#70;&#41;&#47;&#99;&#40;&#65;&#44;&#66;&#41;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"234\" style=\"vertical-align: -6px;\"\/>.<\/figcaption><\/figure><\/div>\n\n\n<p>Solving the geometry problem for the counterclockwise-most subtending angle in this worst-case sitation, we conclude the eigenangle bound <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e666336b5c80d29ed3ef3daf95a0d27e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#45;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#92;&#108;&#101;&#32;&#92;&#115;&#105;&#110;&#94;&#123;&#45;&#49;&#125;&#40;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#47;&#99;&#40;&#65;&#44;&#66;&#41;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"272\" style=\"vertical-align: -6px;\"\/>. An entirely analogous argument using the max-min variational principle (1) proves an identical lower bound, thus showing<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 43px;\"><span class=\"ql-right-eqno\"> (3) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b1b7bc577993964a51beda3ed2ffe2b8_l3.png\" height=\"43\" width=\"195\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#115;&#105;&#110;&#32;&#124;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#45;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#124;&#32;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#125;&#123;&#99;&#40;&#65;&#44;&#66;&#41;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>In the language of eigenvalues, we have<sup class=\"modern-footnotes-footnote \" data-mfn=\"19\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-19\">19<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-19\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"19\">I&#8217;m being a little sloppy here. For a result like this to truly hold, I believe all of the perturbed and unperturbed eigenangles should all be contained in one half of the complex plane.<\/span>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 43px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c4a1b8fe61d0a739091e3751e69562d8_l3.png\" height=\"43\" width=\"359\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#124;&#92;&#99;&#111;&#116;&#94;&#123;&#45;&#49;&#125;&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#125;&#95;&#106;&#41;&#32;&#45;&#32;&#92;&#99;&#111;&#116;&#94;&#123;&#45;&#49;&#125;&#40;&#92;&#108;&#97;&#109;&#98;&#100;&#97;&#95;&#106;&#41;&#124;&#32;&#92;&#108;&#101;&#32;&#92;&#115;&#105;&#110;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#125;&#123;&#99;&#40;&#65;&#44;&#66;&#41;&#125;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<h2 class=\"wp-block-heading\">Interpreting Stewart&#8217;s Theory<\/h2>\n\n\n\n<p>After much work, we have finally proven our first generalized eigenvalue perturbation theorem. After taking a moment to celebrate, let&#8217;s ask ourselves: what does this result tell us? <\/p>\n\n\n\n<p>Let&#8217;s start with the good. This result shows us that if the perturbation, measured by the numerical radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8e29e8ac9155ff52f2d8b3bf61c58b71_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#105;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"77\" style=\"vertical-align: -5px;\"\/>, is much smaller than the definiteness of the original problem, measured by the Crawford number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/>, then the <em>eigenangles<\/em> change by a small amount. What does this mean in terms of the eigenvalues? For small eigenvalues (say, less than one in magnitude), small changes in the eigenangles also lead to small changes of the eigenvalues. However, for large eigenangles, small changes in the eigenangle are magnified into potentially large changes in the eigenvalues. One can view this result in a positive or negative framing. On the one hand, large eigenvalues could be subject to dramatic changes by small perturbations; on the other hand, the small eigenvalues aren&#8217;t &#8220;taken down with the ship&#8221; and are much more well-behaved.<\/p>\n\n\n\n<p>Stewart&#8217;s theory is beautiful. The variational characterization of the eigenangles (1) is a master stroke and exactly the extension one would want from the standard Hermitian\/symmetric theory. From the variational characterization, the perturbation theorem follows almost effortlessly from a little trigonometry. However, Stewart&#8217;s theory has one important deficit: the Crawford number. All that Stewart&#8217;s theory tells is that all of the eigenangles change by at most roughly &#8220;perturbation size over Crawford number&#8221;. If the Crawford number is quite small since the problem is nearly indefinite, this becomes a tough pill to swallow.<\/p>\n\n\n\n<p>The Crawford number is in some ways essential: if the perturbation size exceeds the Crawford number, the problem can become indefinite or even singular. Thus, we have no hope of fully removing the Crawford number from our analysis. But might it be the case that some eigenangles change by much less than &#8220;pertrubation size over Crawford number&#8221;? Could we possibly improve to a result of the form &#8220;the eigenangles change by roughly perturbation size over something (potentially) much less than the Crawford number&#8221;? Sun improved Stewart&#8217;s analysis <a href=\"http:\/\/www.sciencedirect.com\/science\/article\/pii\/0024379582901197\">in 1982<\/a>, but the scourge of the Crawford number remained.<sup class=\"modern-footnotes-footnote \" data-mfn=\"20\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-20\">20<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-20\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"20\">Sun&#8217;s bound does not explicitly have the Crawford number, instead using the quantity <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-49fa2e15d21eaf3751f716b13419d438_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#122;&#101;&#116;&#97;&#32;&#58;&#61;&#32;&#92;&#109;&#97;&#120;&#95;&#123;&#92;&#124;&#120;&#92;&#124;&#61;&#49;&#125;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#124;&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;&#124;&#47;&#124;&#120;&#94;&#42;&#40;&#65;&#43;&#105;&#66;&#41;&#120;&#124;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"354\" style=\"vertical-align: -8px;\"\/> and another hard-to-concisely describe quantity. In many cases, one has nothing better to do than to bound <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-40fa5ccb8911e2874d413c760027cb1f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#122;&#101;&#116;&#97;&#32;&#92;&#108;&#101;&#32;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#47;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"174\" style=\"vertical-align: -5px;\"\/>, in which case the Crawford number has appeared again.<\/span> The theory of Mathias and Li, published in a <a href=\"http:\/\/www.maths.manchester.ac.uk\/ higham\/narep\/narep457.pdf\">technical report in 2004<\/a>, finally produced a bound where the Crawford number is replaced.<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">The Mathias\u2013Li Insight and Reduction to Diagonal Form<\/h2>\n\n\n\n<p>Let&#8217;s go back to the Stewart theory and look for a possible improvement. Recall in the Stewart theory that we considered the point <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> on the complex plane. We then argued that, in the worst case, this point would lie a distance <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> from the origin and then drew a circle around it with radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/>. To improve on Stewart&#8217;s bound, we must somehow do something better than using the fact that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-10b6c3984bd3330ff44901a6fa4a6975_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#124;&#92;&#103;&#101;&#32;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -5px;\"\/>. The insight of the Mathias\u2013Li theory is, in some sense, as simple as this: <em>rather than using the fact that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3a39aa1134c7f71ecb343e3aa4c4ddd9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#124;&#32;&#92;&#103;&#101;&#32;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"184\" style=\"vertical-align: -5px;\"\/> (as in Stewart&#8217;s analysis), use how far <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-eacca144ac0e9a0f4a9a6e16be6540e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"96\" style=\"vertical-align: -5px;\"\/> actually is from zero, where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> is chosen to be the unit norm eigenvectors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>.<\/em><sup class=\"modern-footnotes-footnote \" data-mfn=\"21\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-21\">21<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-21\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"21\">This insight is made more nontrivial by the fact that, in the context of generalized eigenvalue problems, it is often not convenient to choose the eigenvectors to have unit norm. As Mathias and Li note, there are often two more popular normalizations for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/>. If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> is positive definite, one often normalizes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-7a03e667b17c9ee3351f8c17cd521d2a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;&#61;&#32;&#120;&#94;&#42;&#66;&#120;&#32;&#61;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"109\" style=\"vertical-align: -4px;\"\/>\u2014the eigenvectors are thus made &#8220;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>-orthonormal&#8221;, generalizing the fact that the eigenvectors of a Hermitian\/symmetric matrix are orthonormal. Another popular normalization is to scale <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fa757245d7cf462e8237f0b4cded6e9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#43;&#105;&#92;&#98;&#101;&#116;&#97;&#124;&#32;&#61;&#32;&#124;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#124;&#32;&#61;&#32;&#49;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"219\" style=\"vertical-align: -5px;\"\/>. In this way, just taking the eigenvector <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b312d649591164b7149ed0756f694a76_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"10\" style=\"vertical-align: 0px;\"\/> to have unit norm is already a nontrivial insight.<\/span>\n\n\n\n<p>Before going further, let us quickly make a small reduction which will simplify our lives greatly. Letting <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ee8974e4adfbdab75fa43f4df80b4e5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> denote a matrix whose columns are the unit-norm eigenvectors of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/>, one can verify that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-be257e09e1220c45b1ca15b19eaab49f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#94;&#42;&#65;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"53\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-50b90e4cf2935e9dc91ac6bcc5f68722_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;&#94;&#42;&#66;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"54\" style=\"vertical-align: 0px;\"\/> are diagonal matrices with entries <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2b7d328ca274c856ace2c51a7f5dff2f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"77\" style=\"vertical-align: -4px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-68219b2463cd6434f9ec4d54c1b02fdf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#95;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#92;&#98;&#101;&#116;&#97;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"75\" style=\"vertical-align: -4px;\"\/> respectively. With this in mind, it can make our lives a lot easy to just do a change of variables <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5d25949b3025c8cb242ceaa86d2b7afb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#32;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#32;&#88;&#94;&#42;&#65;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -1px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-18f2de6ae290e1d49fe64171f526551b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#32;&#88;&#94;&#42;&#66;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"96\" style=\"vertical-align: -1px;\"\/> (which in turn sends <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-d7ac7e37dd1e8128798a7fe6958b4a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#32;&#88;&#94;&#42;&#69;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"96\" style=\"vertical-align: -1px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5ee99bc393cea3530251d692d5e47228_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;&#32;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#32;&#88;&#94;&#42;&#70;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"95\" style=\"vertical-align: -1px;\"\/>). The change of variables <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5d25949b3025c8cb242ceaa86d2b7afb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;&#32;&#92;&#109;&#97;&#112;&#115;&#116;&#111;&#32;&#88;&#94;&#42;&#65;&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"94\" style=\"vertical-align: -1px;\"\/> is very common in linear algebra and is called a <a href=\"https:\/\/en.wikipedia.org\/wiki\/Matrix_congruence\">congruence transformation<\/a>.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>Perform a change of variables by a congruence transformation with the matrix of eigenvectors.<\/strong><\/p>\n\n\n\n<p>While this change of variables makes our lives a lot easier, we must first worry about how this change of variables might effect the size of the perturbation matrices <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-653ca903a805276f0592e95ae97d308f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#69;&#44;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\"\/>. It turns out this change of variables is not totally benign, but it is not maximally harmful either. Specifically, the spectral radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> can grow by as much as a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75a5652acadcd645b180a972b75a9d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"22\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-22\">22<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-22\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"22\">This is because, in virtue of having unit-norm columns, the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Matrix_norm#Spectral_norm\">spectral norm<\/a> of the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ee8974e4adfbdab75fa43f4df80b4e5d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#88;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\"\/> matrix is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0b2f90eb613f483344f59b960de914fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#124;&#88;&#92;&#124;&#32;&#92;&#108;&#101;&#32;&#92;&#124;&#88;&#92;&#124;&#95;&#123;&#92;&#114;&#109;&#32;&#70;&#125;&#32;&#92;&#108;&#101;&#32;&#92;&#115;&#113;&#114;&#116;&#123;&#110;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -5px;\"\/>. Further, note the following variational characterization of the spectral radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f2d385a824e624675aaaacdf52e7b57a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#61;&#32;&#92;&#109;&#97;&#120;&#95;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#92;&#124;&#32;&#40;&#92;&#99;&#111;&#115;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#41;&#32;&#69;&#32;&#43;&#32;&#40;&#92;&#115;&#105;&#110;&#92;&#116;&#104;&#101;&#116;&#97;&#41;&#32;&#70;&#32;&#92;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"306\" style=\"vertical-align: -5px;\"\/>. Plugging these two facts together yields <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3279e43b22e80c8f42430a4b46424609_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#88;&#94;&#42;&#69;&#88;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#44;&#32;&#88;&#94;&#42;&#70;&#88;&#41;&#32;&#92;&#108;&#101;&#32;&#92;&#124;&#88;&#92;&#124;&#94;&#50;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#92;&#108;&#101;&#32;&#110;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"412\" style=\"vertical-align: -5px;\"\/>.<\/span> This factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75a5652acadcd645b180a972b75a9d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> isn&#8217;t great, but it is much better than if the bound were to degrade by a factor of the condition number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b89c69153b90132b53e3bdee662b09eb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#124;&#88;&#92;&#124;&#92;&#124;&#88;&#94;&#123;&#45;&#49;&#125;&#92;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"82\" style=\"vertical-align: -5px;\"\/>, which can be arbitrarily large.<\/p>\n\n\n\n<p class=\"has-text-align-center\"><strong>This change of variables may increase <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> by at most a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75a5652acadcd645b180a972b75a9d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>.<\/strong><\/p>\n\n\n\n<p><em>From now on, we shall tacitly assume that this change of variables has taken place, with <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> being diagonal and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1653b8134b59e8fb4e789ac004d93957_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#69;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-7bf5d1207baa8be58658ce9d3cf12043_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#70;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> being such that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> is at most a factor <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75a5652acadcd645b180a972b75a9d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/> larger than it was previously. We denote by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-91a35de95e7dc8ae3dd8fea3c8e894ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -6px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-40572baaedd4987b478e586bff7f13e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"16\" style=\"vertical-align: -6px;\"\/> the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>th diagonal element of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/>, which are given by <em><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8c7e368a5be9b2b7dd07e9429e9e51b2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#61;&#32;&#120;&#95;&#106;&#94;&#42;&#65;&#120;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"89\" style=\"vertical-align: -8px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-4adaab763328961579f3984169e602c9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#32;&#61;&#32;&#120;&#95;&#106;&#94;&#42;&#66;&#120;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"89\" style=\"vertical-align: -8px;\"\/><\/em> where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b579cb2878b1f34cbedeb81a01abd17c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#120;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"16\" style=\"vertical-align: -6px;\"\/> is the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>th unit-norm eigenvector<\/em> <\/p>\n\n\n\n<h2 class=\"wp-block-heading\">Mathias and Li&#8217;s Perturbation Theory<\/h2>\n\n\n\n<p>We first assume the perturbation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-653ca903a805276f0592e95ae97d308f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#69;&#44;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"48\" style=\"vertical-align: -5px;\"\/> is smaller than the Crawford number in the sense <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8279147aba308945f984cfd5590ebc94_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#60;&#32;&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"157\" style=\"vertical-align: -5px;\"\/>, which is required to be assured that the perturbed problem <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fda90273726cf242b0edb2bf5db8a316_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#65;&#125;&#44;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#66;&#125;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"47\" style=\"vertical-align: -5px;\"\/> does not lose definiteness. This will be the only place in this analysis where we use the Crawford number.<\/p>\n\n\n\n<p>Draw a circle of radius <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> around <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b645d187911813700e956e63845e0569_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -6px;\"\/>.<\/p>\n\n\n<div class=\"wp-block-image\">\n<figure class=\"aligncenter size-large is-resized\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"860\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/mathias_li-1024x860.png\" alt=\"\" class=\"wp-image-918\" style=\"width:480px\" srcset=\"https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/mathias_li-1024x860.png 1024w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/mathias_li-300x252.png 300w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/mathias_li-768x645.png 768w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/mathias_li-1536x1291.png 1536w, https:\/\/www.ethanepperly.com\/wp-content\/uploads\/2021\/11\/mathias_li-2048x1721.png 2048w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure><\/div>\n\n\n<p>If <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b51163abcaddb4d93bb8a7ebe6fd5641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"14\" style=\"vertical-align: -6px;\"\/> is the associated eigenangle, then this circle is subtended by arcs with angles<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 44px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8dd306bc1e825486444ab8da3c1abbbc_l3.png\" height=\"44\" width=\"489\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#101;&#108;&#108;&#95;&#106;&#32;&#61;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#45;&#32;&#92;&#115;&#105;&#110;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#125;&#123;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;&#32;&#92;&#113;&#117;&#97;&#100;&#32;&#117;&#95;&#106;&#32;&#61;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#115;&#105;&#110;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#125;&#123;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>It would be nice if the perturbed eigenangles <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1a37212666819ac4914d084854cccb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"14\" style=\"vertical-align: -6px;\"\/> were guaranteed to lie in these arcs (i.e., <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5176725bb671ebc3fffc2c10438c7d38_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#95;&#106;&#32;&#92;&#108;&#101;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#92;&#108;&#101;&#32;&#117;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"93\" style=\"vertical-align: -6px;\"\/>). Unfortunately this is not necessarily the case. If one <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b645d187911813700e956e63845e0569_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"61\" style=\"vertical-align: -6px;\"\/> is close to the origin, it will have a large arc which may intersect with other arcs; if this happens, we can&#8217;t guarantee that each perturbed eigenangle will remain within its individual arc. We can still say something though. <\/p>\n\n\n\n<p>What follows is somewhat technical, so let&#8217;s start with the takeaway conclusion: <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1a37212666819ac4914d084854cccb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"14\" style=\"vertical-align: -6px;\"\/> is larger than any <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/> of the lower bounds <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2a7855c522123bf09057ab82c3c36164_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"13\" style=\"vertical-align: -6px;\"\/>. In particular, this means that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-a1a37212666819ac4914d084854cccb6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"24\" width=\"14\" style=\"vertical-align: -6px;\"\/> is larger than the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>th largest of all the lower bounds. That is, if we rearrange the lower bounds <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f83e648e47a694a9fa358d697d9251e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#95;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#92;&#101;&#108;&#108;&#95;&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"69\" style=\"vertical-align: -4px;\"\/> in decreasing order <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-68fa0540cc96542ea6f02444d0b959bf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#95;&#49;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#32;&#92;&#103;&#101;&#32;&#92;&#101;&#108;&#108;&#95;&#50;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#32;&#92;&#103;&#101;&#32;&#92;&#99;&#100;&#111;&#116;&#115;&#32;&#92;&#103;&#101;&#32;&#92;&#101;&#108;&#108;&#95;&#110;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"137\" style=\"vertical-align: -5px;\"\/>, we hace <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-e584771a7009c147d212de2065d87f70_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#92;&#103;&#101;&#32;&#92;&#101;&#108;&#108;&#95;&#106;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"53\" style=\"vertical-align: -8px;\"\/>. An entirely analogous argument will give an upper bound, yielding<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 26px;\"><span class=\"ql-right-eqno\"> (4) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b001ae16887c2ec3eff6839c024d0f5b_l3.png\" height=\"26\" width=\"99\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#101;&#108;&#108;&#95;&#106;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#32;&#92;&#108;&#101;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#92;&#108;&#101;&#32;&#117;&#95;&#106;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>For those interested in the derivation, read on the in the following optional section:<\/p>\n\n\n<div class=\"su-spoiler su-spoiler-style-default su-spoiler-icon-plus su-spoiler-closed\" data-scroll-offset=\"0\" data-anchor-in-url=\"no\"><div class=\"su-spoiler-title\" tabindex=\"0\" role=\"button\"><span class=\"su-spoiler-icon\"><\/span>Derivation of the Mathias\u2013Li Bounds<\/div><div class=\"su-spoiler-content su-u-clearfix su-u-trim\">Since <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are diagonal, the eigenvectors of the pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-bce7cef2561083b562144eb956b3eab3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"47\" style=\"vertical-align: -5px;\"\/> are just the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Standard_basis\">standard basis vectors<\/a>, the <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>th of which we will denote <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-90b0dbac57d4a617137dbf644bf9b4fe_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"14\" style=\"vertical-align: -6px;\"\/>. The trick will be to use the max-min characterization (1) with the subspace <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-ef4ca273d35c61336fca8006b215f82c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"15\" style=\"vertical-align: 0px;\"\/> spanned by some collection of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/> basis vectors <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0a9f803077ad9adaa4251cd33820cc6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#101;&#95;&#123;&#105;&#95;&#49;&#125;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#101;&#95;&#123;&#105;&#95;&#106;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"78\" style=\"vertical-align: -7px;\"\/>. Churning through a couple inequalities in quick fashion,<sup class=\"modern-footnotes-footnote \" data-mfn=\"23\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-23\">23<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-23\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"23\">See pg. 17 of the <a href=\"http:\/\/www.maths.manchester.ac.uk\/ higham\/narep\/narep457.pdf\">Mathias and Li report<\/a>.<\/span> we obtain<\/p>\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 202px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-8b098c1adecc1e3275b29d5fa26ab0cd_l3.png\" height=\"202\" width=\"579\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125; &#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#38;&#92;&#103;&#101;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#92;&#115;&#117;&#98;&#115;&#116;&#97;&#99;&#107;&#123;&#120;&#32;&#92;&#105;&#110;&#32;&#92;&#109;&#97;&#116;&#104;&#99;&#97;&#108;&#123;&#88;&#125;&#32;&#92;&#92;&#32;&#92;&#124;&#120;&#92;&#124;&#32;&#61;&#32;&#49;&#125;&#125;&#32;&#92;&#97;&#114;&#103;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#32;&#120;&#94;&#42;&#40;&#65;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#66;&#41;&#120;&#32;&#43;&#32;&#120;&#94;&#42;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#120;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#32;&#92;&#92; &#38;&#92;&#103;&#101;&#32;&#92;&#109;&#105;&#110;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#32;&#92;&#97;&#114;&#103;&#40;&#121;&#43;&#122;&#41;&#32;&#58;&#32;&#121;&#32;&#92;&#105;&#110;&#32;&#92;&#111;&#112;&#101;&#114;&#97;&#116;&#111;&#114;&#110;&#97;&#109;&#101;&#123;&#99;&#111;&#110;&#118;&#125;&#32;&#92;&#123;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#123;&#105;&#95;&#49;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#123;&#105;&#95;&#49;&#125;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#123;&#105;&#95;&#106;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#123;&#105;&#95;&#106;&#125;&#32;&#92;&#125;&#44;&#32;&#122;&#92;&#105;&#110;&#32;&#87;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#92;&#125;&#32;&#92;&#92; &#38;&#92;&#103;&#101;&#32;&#92;&#109;&#105;&#110;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#32;&#92;&#97;&#114;&#103;&#40;&#121;&#43;&#122;&#41;&#32;&#58;&#32;&#121;&#32;&#92;&#105;&#110;&#32;&#92;&#111;&#112;&#101;&#114;&#97;&#116;&#111;&#114;&#110;&#97;&#109;&#101;&#123;&#99;&#111;&#110;&#118;&#125;&#32;&#92;&#123;&#32;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#123;&#105;&#95;&#49;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#123;&#105;&#95;&#49;&#125;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#123;&#105;&#95;&#106;&#125;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#123;&#105;&#95;&#106;&#125;&#32;&#92;&#125;&#44;&#32;&#124;&#122;&#124;&#92;&#108;&#101;&#32;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#92;&#125;&#32;&#92;&#92; &#38;&#92;&#103;&#101;&#32;&#92;&#109;&#105;&#110;&#32;&#92;&#108;&#101;&#102;&#116;&#92;&#123;&#32;&#92;&#97;&#114;&#103;&#40;&#119;&#41;&#32;&#58;&#32;&#119;&#92;&#105;&#110;&#92;&#111;&#112;&#101;&#114;&#97;&#116;&#111;&#114;&#110;&#97;&#109;&#101;&#123;&#99;&#111;&#110;&#118;&#125;&#32;&#92;&#98;&#105;&#103;&#99;&#117;&#112;&#95;&#123;&#107;&#61;&#49;&#125;&#94;&#106;&#32;&#92;&#123;&#32;&#97;&#92;&#105;&#110;&#92;&#109;&#97;&#116;&#104;&#98;&#98;&#123;&#67;&#125;&#32;&#58;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#123;&#105;&#95;&#107;&#125;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#123;&#105;&#95;&#107;&#125;&#32;&#45;&#32;&#97;&#124;&#32;&#92;&#108;&#101;&#32;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#32;&#92;&#125;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#92;&#125;&#32;&#92;&#92; &#38;&#61;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#107;&#61;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#106;&#125;&#32;&#92;&#101;&#108;&#108;&#95;&#123;&#105;&#95;&#107;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#97;&#108;&#105;&#103;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n<p>Here, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-67c9aadf3fcfaedfbed91a8dbd7f2424_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#111;&#112;&#101;&#114;&#97;&#116;&#111;&#114;&#110;&#97;&#109;&#101;&#123;&#99;&#111;&#110;&#118;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"35\" style=\"vertical-align: 0px;\"\/> denotes the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Convex_hull\">convex hull<\/a>. Since this holds for every set of indices <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5ab40f04d3b1a236d225b0654369c974_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#105;&#95;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#105;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"65\" style=\"vertical-align: -6px;\"\/>, it in particular holds for the set of indices which makes <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b124d6aa0ffe1963c2eb0f691dc8a0a3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#105;&#110;&#95;&#123;&#107;&#61;&#49;&#44;&#92;&#108;&#100;&#111;&#116;&#115;&#44;&#106;&#125;&#32;&#92;&#101;&#108;&#108;&#95;&#123;&#105;&#95;&#107;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"102\" style=\"vertical-align: -6px;\"\/> the largest. Thus, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-c903ab4e3e8fe7927ce02b34b5bc6ecd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#92;&#103;&#101;&#32;&#92;&#101;&#108;&#108;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"53\" style=\"vertical-align: -8px;\"\/>.<\/div><\/div>\n\n\n<h2 class=\"wp-block-heading\">How to Use Mathias\u2013Li&#8217;s Perturbation Theory<\/h2>\n\n\n\n<p>The eigenangle perturbation bound (4) can be instantiated in a variety of ways. We briefly sketch two. The first is to bound <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-af6732aede701ef337b071585557db68_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"68\" style=\"vertical-align: -6px;\"\/> by its minimum over all <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-fe087f8cefab0bcb3270609914ada26c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"9\" style=\"vertical-align: -4px;\"\/>, which then gives a bound on <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-6a467d6d353057189377e62554e05c3d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"17\" style=\"vertical-align: -8px;\"\/> (and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-85d49eed43c5a0cadc3fe8cf2d5fdeed_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"14\" style=\"vertical-align: -8px;\"\/>)<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 44px;\"><span class=\"ql-right-eqno\"> &nbsp; <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1c21fa2a8a4c546aa4f39e267af13438_l3.png\" height=\"44\" width=\"538\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#32;&#92;&#103;&#101;&#32;&#92;&#109;&#105;&#110;&#95;&#123;&#49;&#92;&#108;&#101;&#32;&#106;&#92;&#108;&#101;&#32;&#110;&#125;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#32;&#92;&#105;&#109;&#112;&#108;&#105;&#101;&#115;&#32;&#117;&#95;&#106;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#32;&#92;&#108;&#101;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#115;&#105;&#110;&#94;&#123;&#45;&#49;&#125;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#125;&#123;&#92;&#109;&#105;&#110;&#95;&#123;&#49;&#92;&#108;&#101;&#32;&#106;&#92;&#108;&#101;&#32;&#110;&#125;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>Plugging into (4) and simplifying gives<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 43px;\"><span class=\"ql-right-eqno\"> (5) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-6c353ca8f458f5caa66fb2a0af4784b2_l3.png\" height=\"43\" width=\"269\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#115;&#105;&#110;&#32;&#92;&#108;&#101;&#102;&#116;&#124;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#45;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#124;&#32;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#125;&#123;&#92;&#109;&#105;&#110;&#95;&#123;&#49;&#92;&#108;&#101;&#32;&#106;&#92;&#108;&#101;&#32;&#110;&#125;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>This improves on Stewart&#8217;s bound (3) by replacing the Crawford number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5d6661a3a3924245d2eb2b5c8d3f4240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#105;&#110;&#95;&#123;&#49;&#92;&#108;&#101;&#32;&#106;&#92;&#108;&#101;&#32;&#110;&#125;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -6px;\"\/>; as Mathias and Li show <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5d6661a3a3924245d2eb2b5c8d3f4240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#105;&#110;&#95;&#123;&#49;&#92;&#108;&#101;&#32;&#106;&#92;&#108;&#101;&#32;&#110;&#125;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -6px;\"\/> is always smaller than or equal to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> and can be <em>much much smaller<\/em>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"24\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-24\">24<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-24\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"24\">Recall that Mathias and Li&#8217;s bound first requires us to do a change of variables where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> both become diagonal, which can increase <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-1a5dc115aaefd1e5726a85761745b325_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"76\" style=\"vertical-align: -5px;\"\/> by a factor of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75a5652acadcd645b180a972b75a9d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>. Thus, for an apples-to-apples comparison with Stewart&#8217;s theory where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> are non-diagonal, (5) should be interpreted as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-929db08553b62e877dfa025b0a9c049e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#115;&#105;&#110;&#32;&#92;&#108;&#101;&#102;&#116;&#124;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#45;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#124;&#32;&#92;&#108;&#101;&#32;&#110;&#92;&#44;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#47;&#92;&#109;&#105;&#110;&#123;&#49;&#92;&#108;&#101;&#32;&#106;&#92;&#108;&#101;&#32;&#110;&#125;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"393\" style=\"vertical-align: -12px;\"\/>.<\/span>\n\n\n\n<p>For the second instantiation (4), we recognize that if an eigenangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b51163abcaddb4d93bb8a7ebe6fd5641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"14\" style=\"vertical-align: -6px;\"\/> is sufficiently well-separated from other eigenangles (relative to the size of the perturbation and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5d6661a3a3924245d2eb2b5c8d3f4240_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#109;&#105;&#110;&#95;&#123;&#49;&#92;&#108;&#101;&#32;&#106;&#92;&#108;&#101;&#32;&#110;&#125;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"146\" style=\"vertical-align: -6px;\"\/>), then we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-5f0de6f145238d9eaaec75641c33bafa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#117;&#95;&#106;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#32;&#92;&#108;&#101;&#32;&#117;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"57\" style=\"vertical-align: -8px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f83e49a4d1d1388e6b816e7847285d80_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#101;&#108;&#108;&#95;&#106;&#94;&#92;&#100;&#111;&#119;&#110;&#97;&#114;&#114;&#111;&#119;&#32;&#92;&#103;&#101;&#32;&#92;&#101;&#108;&#108;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"52\" style=\"vertical-align: -8px;\"\/>. (The precise instantiation of &#8220;sufficiently well-separated&#8221; requires some tedious algebra; if you&#8217;re interested, see <a href=\"http:\/\/www.maths.manchester.ac.uk\/ higham\/narep\/narep457.pdf\">Footnote 7 in Mathias and Li&#8217;s paper<\/a>.<sup class=\"modern-footnotes-footnote \" data-mfn=\"25\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-25\">25<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-25\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"25\">You may also be interested in <a href=\"https:\/\/arxiv.org\/pdf\/2110.07492.pdf\">Corollary 2.2 in this preprint<\/a> by myself and coauthors.<\/span>) Under this separation condition, (4) then reduces to<\/p>\n\n\n<p><p class=\"ql-center-displayed-equation\" style=\"line-height: 43px;\"><span class=\"ql-right-eqno\"> (6) <\/span><span class=\"ql-left-eqno\"> &nbsp; <\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-24a968603bcf8271e254e5119fbaee1e_l3.png\" height=\"43\" width=\"197\" class=\"ql-img-displayed-equation quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125; &#92;&#115;&#105;&#110;&#32;&#92;&#108;&#101;&#102;&#116;&#124;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#45;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#124;&#32;&#92;&#108;&#101;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#114;&#40;&#69;&#43;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#70;&#41;&#125;&#123;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#125;&#46; &#92;&#101;&#110;&#100;&#123;&#101;&#113;&#117;&#97;&#116;&#105;&#111;&#110;&#42;&#125;\" title=\"Rendered by QuickLaTeX.com\"\/><\/p><\/p>\n\n\n<p>This result improves on Stewart&#8217;s result (4) by even more, since we have now replaced the Crawford number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-9fa234665f0bea538036baf2c00f609a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#99;&#40;&#65;&#44;&#66;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"56\" style=\"vertical-align: -5px;\"\/> by <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-3f735820ba9fc2a71d3755aa57df7cb3_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"68\" style=\"vertical-align: -6px;\"\/> for a sufficiently small perturbation. In fact, a result of this form is nearly as good as one could hope for.<sup class=\"modern-footnotes-footnote \" data-mfn=\"26\" data-mfn-post-scope=\"000000000000057e0000000000000000_808\"><a href=\"javascript:void(0)\"  role=\"button\" aria-pressed=\"false\" aria-describedby=\"mfn-content-000000000000057e0000000000000000_808-26\">26<\/a><\/sup><span id=\"mfn-content-000000000000057e0000000000000000_808-26\" role=\"tooltip\" class=\"modern-footnotes-footnote__note\" tabindex=\"0\" data-mfn=\"26\">Specifically, the <a href=\"https:\/\/en.wikipedia.org\/wiki\/Condition_number\">condition number<\/a> of the eigenangle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-b51163abcaddb4d93bb8a7ebe6fd5641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"14\" style=\"vertical-align: -6px;\"\/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f92e83446886630d35e990203956afec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"87\" style=\"vertical-align: -6px;\"\/>, so we know for sufficiently small perturbations we have <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75695e87cc89b9d5e967fc405721c48a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#124;&#32;&#92;&#119;&#105;&#100;&#101;&#116;&#105;&#108;&#100;&#101;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#125;&#95;&#106;&#32;&#45;&#32;&#92;&#116;&#104;&#101;&#116;&#97;&#95;&#106;&#32;&#92;&#114;&#105;&#103;&#104;&#116;&#124;&#32;&#92;&#108;&#101;&#115;&#115;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#40;&#92;&#109;&#98;&#111;&#120;&#123;&#115;&#105;&#122;&#101;&#32;&#111;&#102;&#32;&#112;&#101;&#114;&#116;&#117;&#114;&#98;&#97;&#116;&#105;&#111;&#110;&#125;&#41;&#32;&#92;&#116;&#105;&#109;&#101;&#115;&#32;&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"362\" style=\"vertical-align: -12px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-f92e83446886630d35e990203956afec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#124;&#92;&#97;&#108;&#112;&#104;&#97;&#95;&#106;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#32;&#92;&#98;&#101;&#116;&#97;&#95;&#106;&#124;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"87\" style=\"vertical-align: -6px;\"\/> is the smallest number for which such a relation holds. Mathias and Li&#8217;s theory allows for a statement of this form to be made rigorous for a finite-size perturbation. Again, the only small deficit is the additional factor of &#8220;<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-75a5652acadcd645b180a972b75a9d09_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\"\/>&#8221; from the change of variables to diagonal form.<\/span>\n\n\n\n<h2 class=\"wp-block-heading\">The Elegant Geometry of Generalized Eigenvalue Perturbation Theory<\/h2>\n\n\n\n<p>As I said at the start of this post, what fascinates me about this generalized eigenvalue perturbation is the beautiful and elegant geometry. When I saw it for the first time, it felt like a magic trick: a definite generalized eigenvalue problem with real eigenvalues was transformed by sleight of hand into a geometry problem on the complex plane, with solutions involving just a little high school geometry and trigonometry. Upon studying the theory, I began to appreciate it for a different reason. Upon closer examination, the magic trick was revealed to be a sequence of deductions, each logically following naturally from the last. To the pioneers of this subject\u2014Stewart, Sun, Mathias, Li, and others\u2014this sequence of logical deductions was not preordained, and their discovery of this theory doubtlessly required careful thought and leaps of insight. Now that this theory has been discovered, however, we get the benefit of retrospection, and can retell a narrative of this theory where each step follows naturally from the last. When told this way, one almost imagines being able to develop this theory by oneself, where at each stage we appeal to some notion of mathematical elegance (e.g., by treating <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-11f4f587954b361e7d78940f65b8d70d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#65;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"13\" style=\"vertical-align: 0px;\"\/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-06d2c1a5a171b6d7d9c5df87d123c5a4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#66;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\"\/> symmetrically) or by applying a standard trick (e.g., identifying a pair <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-2fb35cffe2c4eadf60573bc2c613b0af_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#40;&#92;&#97;&#108;&#112;&#104;&#97;&#44;&#92;&#98;&#101;&#116;&#97;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"42\" style=\"vertical-align: -5px;\"\/> with the complex number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.ethanepperly.com\/wp-content\/ql-cache\/quicklatex.com-0471a28c0fc54a660cc5346fa4efd92c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#108;&#112;&#104;&#97;&#32;&#43;&#32;&#92;&#109;&#97;&#116;&#104;&#114;&#109;&#123;&#105;&#125;&#92;&#98;&#101;&#116;&#97;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"49\" style=\"vertical-align: -4px;\"\/>). Since this theory took several decades to fall into place, we should not let this storytelling exercise fool us into thinking the prospective act of developing a new theory will be as straightforward and linear as this retelling, pruned of dead ends and halts in progress, might suggest. <\/p>\n\n\n\n<p>That said, I do think the development of the perturbation theory of the generalized eigenvalue problem does have a lesson for those of us who seek to develop mathematical theories: be guided by mathematical elegance. At several points in the development of the perturbation theory, we obtained great gains by treating quantities which play a symmetric role in the problem symmetrically in the theory or by treating a pair of real numbers as a complex number and asking how to interpret that complex number. My hope is that this perturbation theory serves as a good example for how letting oneself be guided by intuition, a small array of standard tricks, and a search for elegance can lead one to conceptualize a problem in the right way which leads (after a considerable amount of effort and a few lucky breaks) to a natural solution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Unfortunately, Mathias and Li&#8217;s original paper\u2014on which this blog post is based\u2014appears to have been taken offline. I am uploading a copy here for reference: In this post, I want to discuss a beautiful and simple geometric picture of the perturbation theory of definite generalized eigenvalue problems. As a culmination, we&#8217;ll see a taste of<a class=\"more-link\" href=\"https:\/\/www.ethanepperly.com\/index.php\/2021\/12\/14\/the-elegant-geometry-of-the-generalized-eigenvalue-perturbation-theory\/\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[6],"tags":[],"class_list":["post-808","post","type-post","status-publish","format-standard","hentry","category-expository"],"_links":{"self":[{"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/posts\/808","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/comments?post=808"}],"version-history":[{"count":81,"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/posts\/808\/revisions"}],"predecessor-version":[{"id":2240,"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/posts\/808\/revisions\/2240"}],"wp:attachment":[{"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/media?parent=808"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/categories?post=808"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.ethanepperly.com\/index.php\/wp-json\/wp\/v2\/tags?post=808"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}