{"id":64900,"date":"2025-06-20T10:30:57","date_gmt":"2025-06-20T07:30:57","guid":{"rendered":"https:\/\/wpcalc.com\/en\/?p=64900"},"modified":"2025-06-20T10:30:57","modified_gmt":"2025-06-20T07:30:57","slug":"eigenvalues-eigenvectors-3x3","status":"publish","type":"post","link":"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-3x3\/","title":{"rendered":"Eigenvalues and Eigenvectors 3\u00d73 Matrix Calculator"},"content":{"rendered":"<section class=\"not-prose calculator-box\" aria-labelledby=\"calculator-title\"><div class=\"flex items-baseline sm:items-center justify-between gap-2 sm:gap-3\"><div class=\"flex flex-col sm:flex-row sm:items-center gap-2\"><span class=\"icon icon-calculator text-primary-700! text-base! dark:text-primary-300!\" aria-hidden=\"true\"><\/span><h2 class=\"text-lg font-display font-bold\">3\u00d73 Matrix Eigenvalues and Eigenvectors Solver<\/h2><\/div><div class=\"relative group inline-block\">\n  <button class=\"favorite\" id=\"favorite\" data-favorite-id=\"64900\" data-favorite-title=\"Eigenvalues and Eigenvectors 3\u00d73\" data-favorite-url=\"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-3x3\/\" data-favorite-excerpt=\"This calculator finds the eigenvalues and eigenvectors of a 3\u00d73 matrix. It\u2019s ideal for students, engineers, and scientists working with linear algebra, physics, and advanced mathematics where matrix diagonalization and spectral analysis are required.\" aria-label=\"Add to Favorites\" data-favorite-icon=\"icon icon-binary\">\n    <span class=\"icon icon-shape-star-empty\"><\/span>\n  <\/button>\n  <div class=\"absolute right-full -translate-y-1\/2 top-1\/2 mr-2 w-max max-w-xs px-3 py-2 bg-gray-800 text-white text-xs rounded shadow-lg opacity-0 group-hover:opacity-100 transition-opacity duration-200 z-10 pointer-events-none\">\n   <span class=\"favorite-tooltip\" id=\"favorite-tooltip\"><\/span>\n  <\/div>\n<\/div><\/div><form action=\"https:\/\/wpcalc.com\/en\/mathematics\/eigenvalues-eigenvectors-3x3\/\" method=\"POST\" class=\"calculator\" id=\"calculator-64900\" data-post=\"64900\"><fieldset class=\"fieldset-input is-3\"><legend class=\"sr-only\">Input Fields<\/legend><div class=\"field has-term\" id=\"input-1\"><label for=\"field-1\">a\u2081\u2081<\/label><div class=\"term\">a11<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-1\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-2\"><label for=\"field-2\">a\u2081\u2082<\/label><div class=\"term\">a12<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-2\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-3\"><label for=\"field-3\">a\u2081\u2083<\/label><div class=\"term\">a13<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-3\" step=\"any\" value=\"3\"\/><\/div><div class=\"field has-term\" id=\"input-4\"><label for=\"field-4\">a\u2082\u2081<\/label><div class=\"term\">a21<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-4\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-5\"><label for=\"field-5\">a\u2082\u2082<\/label><div class=\"term\">a22<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-5\" step=\"any\" value=\"4\"\/><\/div><div class=\"field has-term\" id=\"input-6\"><label for=\"field-6\">a\u2082\u2083<\/label><div class=\"term\">a23<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-6\" step=\"any\" value=\"2\"\/><\/div><div class=\"field has-term\" id=\"input-7\"><label for=\"field-7\">a\u2083\u2081<\/label><div class=\"term\">a31<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-7\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-8\"><label for=\"field-8\">a\u2083\u2082<\/label><div class=\"term\">a32<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-8\" step=\"any\" value=\"1\"\/><\/div><div class=\"field has-term\" id=\"input-9\"><label for=\"field-9\">a\u2083\u2083<\/label><div class=\"term\">a33<\/div>  <div class=\"absolute bottom-0 left-16 top-8 w-2 h-0.5 bg-blue-500\"><\/div><input type=\"number\" name=\"a\" id=\"field-9\" step=\"any\" value=\"3\"\/><\/div><\/fieldset><div class=\"buttons\"><button type=\"submit\" data-text=\"Re-Calculate\" id=\"calculate-button\" data-post=\"64900\">Calculate<\/button><button type=\"reset\">Reset<\/button><\/div><div class=\"field is-checkbox hidden!\" id=\"field-auto-calc\"><input type=\"checkbox\" id=\"auto-calc\"><label for=\"auto-calc\">Calculate automatically<\/label><small>If enabled, the result will update automatically when you change any value.<\/small><\/div><div class=\"fieldset-result is-hidden\" aria-labelledby=\"result-title\" aria-live=\"polite\" role=\"region\"> <h3 class=\"result-title bg-gradient-to-r from-primary-50 to-gray-50 dark:from-primary-900 dark:to-gray-800\" id=\"result-title\"><span class=\"icon icon-s-pulse\" aria-hidden=\"true\"><\/span> Your Results<\/h3><div class=\"result-box\"><div class=\"field-result\" id=\"output-1\"><span class=\"field-title\"><span>Determinant |A|<\/span><\/span><span class=\"field-value\" id=\"result-1\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-2\"><span class=\"field-title\"><span>Trace of matrix<\/span><\/span><span class=\"field-value\" id=\"result-2\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-3\"><span class=\"field-title\"><span>Characteristic polynomial |A - cI|<\/span><\/span><span class=\"field-value\" id=\"result-3\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-4\"><span class=\"field-title\"><span>Eigenvalue \u03bb\u2081<\/span><\/span><span class=\"field-value\" id=\"result-4\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-5\"><span class=\"field-title\"><span>Eigenvalue \u03bb\u2082<\/span><\/span><span class=\"field-value\" id=\"result-5\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-6\"><span class=\"field-title\"><span>Eigenvalue \u03bb\u2083<\/span><\/span><span class=\"field-value\" id=\"result-6\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-7\"><span class=\"field-title\"><span>Eigenvector for \u03bb\u2081<\/span><\/span><span class=\"field-value\" id=\"result-7\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-8\"><span class=\"field-title\"><span>Eigenvector for \u03bb\u2082<\/span><\/span><span class=\"field-value\" id=\"result-8\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><div class=\"field-result\" id=\"output-9\"><span class=\"field-title\"><span>Eigenvector for \u03bb\u2083<\/span><\/span><span class=\"field-value\" id=\"result-9\"><\/span><button class=\"copy-result\" data-tooltip=\"Copy Result\"><span class=\"copy-icon icon icon-document-copy\"><\/span><\/button><\/div><\/div><\/div><a href=\"#respond\" class=\"hidden transition-opacity duration-300 opacity-0 w-50 text-sm justify-center items-center gap-2 px-4 py-2 rounded bg-gray-200 text-gray-700 hover:bg-gray-300\" id=\"leave-comment\"><span class=\"icon icon-comments\" aria-hidden=\"true\"><\/span>Leave a Comment<\/a><\/form><\/section><section id=\"calc-reactions\" class=\"not-prose hidden my-12 bg-gradient-to-r from-primary-50 to-gray-50 border border-indigo-100 rounded-xl px-6 py-4 shadow-sm\" data-post=\"64900\" aria-live=\"polite\"><h2 class=\"text-sm text-gray-500 text-center mb-2\">How did this result make you feel?<\/h2><div class=\"grid grid-cols-3 sm:flex sm:flex-row sm:justify-around gap-4 sm:items-center sm:flex-wrap\"><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"like\"><span class=\"reaction-count\" data-reaction-count=\"like\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude00 <\/span><span class=\"reaction-description font-medium\">Like <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"helpful\"><span class=\"reaction-count\" data-reaction-count=\"helpful\">0<\/span><span class=\"reaction text-3xl\">\ud83d\udca1 <\/span><span class=\"reaction-description font-medium\">Helpful <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"confused\"><span class=\"reaction-count\" data-reaction-count=\"confused\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude15 <\/span><span class=\"reaction-description font-medium\">Confused <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"disappointed\"><span class=\"reaction-count\" data-reaction-count=\"disappointed\">0<\/span><span class=\"reaction text-3xl\">\ud83d\ude1e <\/span><span class=\"reaction-description font-medium\">Disappointed <\/span><\/button><button class=\"reaction-btn flex flex-col items-center gap-1 text-sm text-gray-700 hover:text-primary-600 cursor-pointer transition-transform hover:scale-105\" data-reaction=\"inaccurate\"><span class=\"reaction-count\" data-reaction-count=\"inaccurate\">0<\/span><span class=\"reaction text-3xl\">\u274c <\/span><span class=\"reaction-description font-medium\">Inaccurate <\/span><\/button><\/div><div id=\"reaction-message\" class=\"hidden mt-8 rounded-md border border-primary-100 bg-white\/50 backdrop-blur-sm px-4 py-2 text-sm text-center shadow-sm\"><\/div><\/section><ins class=\"adsbygoogle\"\n     style=\"display:block; text-align:center; margin: 32px 0;\"\n     data-ad-layout=\"in-article\"\n     data-ad-format=\"fluid\"\n     data-ad-client=\"ca-pub-1721569815777345\"\n     data-ad-slot=\"6317458308\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n<section class=\"formula-box\" aria-labelledby=\"formula-title\">\r\n\t<div id=\"formula-title\" class=\"not-prose font-display mb-4\"><span class=\"icon icon-formula\" aria-hidden=\"true\"><\/span><h2>Eigenvalues and Eigenvectors Formula for 3\u00d73 Matrix<\/h2><\/div>\r\n\t<figure class=\"not-prose formula\">\r\n\t\t<figcaption class=\"formula-title\">Formula<\/figcaption>\r\n\t\t<div class=\"text-base text-gray-800\" id=\"formula\">\n$$\n\\det(A &#8211; \\lambda I) = 0 \\\\[1em]\n(A &#8211; \\lambda I) \\vec{v} = 0\n$$\n<\/div>\r\n\t<\/figure>\r\n\t<br \/>\n<strong>Where:<\/strong><\/p>\n<ul>\n<li>$$A$$ is the 3\u00d73 matrix<\/li>\n<li>$$\\lambda$$ represents an eigenvalue<\/li>\n<li>$$I$$ is the identity matrix<\/li>\n<li>$$\\vec{v}$$ is the corresponding eigenvector<\/li>\n<\/ul>\n<p>The characteristic equation $$\\det(A &#8211; \\lambda I) = 0$$ is solved for $$\\lambda$$, and the resulting $$\\lambda$$ values are used to compute the eigenvectors by solving the homogeneous system.<br \/>\r\n<\/section>\n<section id=\"calculation\" class=\"calculation-box\" aria-labelledby=\"calculation-title\">\r\n\t<div id=\"calculation-title\" class=\"not-prose font-display\"><span class=\"icon icon-unordered-list\" aria-hidden=\"true\"><\/span><h2>Eigenvalues and Eigenvectors 3\u00d73 \u2013 Calculation Example<\/h2><\/div>\r\n\r\n<\/p>\n<p>$$A = \\begin{bmatrix} 6 &#038; 2 &#038; 1 \\\\ 2 &#038; 3 &#038; 1 \\\\ 1 &#038; 1 &#038; 1 \\end{bmatrix} $$<\/p>\n<p>Solve:<br \/>\n$$\\det(A &#8211; \\lambda I) = 0 $$<\/p>\n<p>Characteristic polynomial:<br \/>\n$$\\lambda^3 &#8211; 10 \\lambda^2 + 27 \\lambda &#8211; 18 = 0 $$<\/p>\n<p>Solutions (eigenvalues):<br \/>\n$$\\lambda_1 = 1, \\quad \\lambda_2 = 3, \\quad \\lambda_3 = 6$$<\/p>\n<p>Find eigenvectors by solving:<br \/>\n$$(A &#8211; \\lambda I)\\vec{v} = 0$$<\/p>\n<p>for each \\(\\lambda\\).<\/p>\n\r\n<\/section>\n<p>Eigenvalues and eigenvectors are key in linear transformations, stability analysis, vibration modes, and quantum mechanics. Our calculator simplifies the process of solving the characteristic equation and finding eigenvectors for any 3\u00d73 matrix.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This calculator finds the eigenvalues and eigenvectors of a 3\u00d73 matrix. It\u2019s ideal for students, engineers, and scientists working with linear algebra, physics, and advanced mathematics where matrix diagonalization and spectral analysis are required.<\/p>\n","protected":false},"author":3168,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[7],"tags":[69],"class_list":["post-64900","post","type-post","status-publish","format-standard","hentry","category-mathematics","tag-matrix"],"acf":[],"_links":{"self":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64900","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/users\/3168"}],"replies":[{"embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/comments?post=64900"}],"version-history":[{"count":2,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64900\/revisions"}],"predecessor-version":[{"id":64902,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/posts\/64900\/revisions\/64902"}],"wp:attachment":[{"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/media?parent=64900"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/categories?post=64900"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/wpcalc.com\/en\/wp-json\/wp\/v2\/tags?post=64900"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}