CINXE.COM
Schur decomposition - Wikipedia
<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Schur decomposition - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-sticky-header-enabled vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"1036263b-e37c-426d-941c-6da77c429ede","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Schur_decomposition","wgTitle":"Schur decomposition","wgCurRevisionId":1269572295,"wgRevisionId":1269572295,"wgArticleId":372514,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["Articles with short description","Short description is different from Wikidata","Articles containing proofs","Matrix theory","Matrix decompositions","Issai Schur"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Schur_decomposition","wgRelevantArticleId":372514,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgRedirectedFrom":"Schur_form", "wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":10000,"wgInternalRedirectTargetUrl":"/wiki/Schur_decomposition","wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1064218","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled" :false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["mediawiki.action.view.redirect","ext.cite.ux-enhancements","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","ext.popups","ext.visualEditor.desktopArticleTarget.init","ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging", "ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.16"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Schur decomposition - Wikipedia"> <meta property="og:type" content="website"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Schur_decomposition"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Schur_decomposition&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Schur_decomposition"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Schur_decomposition rootpage-Schur_decomposition skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" title="Main menu" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li><li id="n-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages"><span>Special pages</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Schur+decomposition" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Schur+decomposition" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Schur+decomposition" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Schur+decomposition" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"><!-- CentralNotice --></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Statement" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Statement"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Statement</span> </div> </a> <ul id="toc-Statement-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Proof" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Proof"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Proof</span> </div> </a> <ul id="toc-Proof-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Computation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Computation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Computation</span> </div> </a> <ul id="toc-Computation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Generalized_Schur_decomposition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Generalized_Schur_decomposition"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Generalized Schur decomposition</span> </div> </a> <ul id="toc-Generalized_Schur_decomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Schur decomposition</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 14 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-14" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">14 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Descomposici%C3%B3_de_Schur" title="Descomposició de Schur – Catalan" lang="ca" hreflang="ca" data-title="Descomposició de Schur" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Schur%C5%AFv_rozklad" title="Schurův rozklad – Czech" lang="cs" hreflang="cs" data-title="Schurův rozklad" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Schur-Zerlegung" title="Schur-Zerlegung – German" lang="de" hreflang="de" data-title="Schur-Zerlegung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Factorizaci%C3%B3n_de_Schur" title="Factorización de Schur – Spanish" lang="es" hreflang="es" data-title="Factorización de Schur" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/D%C3%A9composition_de_Schur" title="Décomposition de Schur – French" lang="fr" hreflang="fr" data-title="Décomposition de Schur" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%8A%88%EC%96%B4_%EB%B6%84%ED%95%B4" title="슈어 분해 – Korean" lang="ko" hreflang="ko" data-title="슈어 분해" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Decomposizione_di_Schur" title="Decomposizione di Schur – Italian" lang="it" hreflang="it" data-title="Decomposizione di Schur" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A4%D7%99%D7%A8%D7%95%D7%A7_%D7%A9%D7%95%D7%A8" title="פירוק שור – Hebrew" lang="he" hreflang="he" data-title="פירוק שור" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Schur-felbont%C3%A1s" title="Schur-felbontás – Hungarian" lang="hu" hreflang="hu" data-title="Schur-felbontás" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Stelling_van_Schur" title="Stelling van Schur – Dutch" lang="nl" hreflang="nl" data-title="Stelling van Schur" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B7%E3%83%A5%E3%83%BC%E3%83%AB%E5%88%86%E8%A7%A3" title="シュール分解 – Japanese" lang="ja" hreflang="ja" data-title="シュール分解" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%B0%D0%B7%D0%BB%D0%BE%D0%B6%D0%B5%D0%BD%D0%B8%D0%B5_%D0%A8%D1%83%D1%80%D0%B0" title="Разложение Шура – Russian" lang="ru" hreflang="ru" data-title="Разложение Шура" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BA%D0%BB%D0%B0%D0%B4_%D0%A8%D1%83%D1%80%D0%B0" title="Розклад Шура – Ukrainian" lang="uk" hreflang="uk" data-title="Розклад Шура" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%88%92%E5%B0%94%E5%88%86%E8%A7%A3" title="舒尔分解 – Chinese" lang="zh" hreflang="zh" data-title="舒尔分解" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1064218#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Schur_decomposition" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Schur_decomposition" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Schur_decomposition"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Schur_decomposition&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Schur_decomposition&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Schur_decomposition"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Schur_decomposition&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Schur_decomposition&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Schur_decomposition" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Schur_decomposition" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Schur_decomposition&oldid=1269572295" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Schur_decomposition&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Schur_decomposition&id=1269572295&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FSchur_decomposition"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FSchur_decomposition"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Schur_decomposition&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Schur_decomposition&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1064218" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Schur_form&redirect=no" class="mw-redirect" title="Schur form">Schur form</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Matrix factorisation in mathematics</div> <p>In the <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> discipline of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, the <b>Schur decomposition</b> or <b>Schur triangulation</b>, named after <a href="/wiki/Issai_Schur" title="Issai Schur">Issai Schur</a>, is a <a href="/wiki/Matrix_decomposition" title="Matrix decomposition">matrix decomposition</a>. It allows one to write an arbitrary complex square matrix as <a href="/wiki/Matrix_similarity" title="Matrix similarity">unitarily similar</a> to an <a href="/wiki/Upper-triangular_matrix" class="mw-redirect" title="Upper-triangular matrix">upper triangular matrix</a> whose diagonal elements are the eigenvalues of the original matrix. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schur_decomposition&action=edit&section=1" title="Edit section: Statement"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The complex Schur decomposition reads as follows: if <span class="texhtml mvar" style="font-style:italic;">A</span> is an <span class="texhtml"><i>n</i> × <i>n</i></span> <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> with <a href="/wiki/Complex_numbers" class="mw-redirect" title="Complex numbers">complex</a> entries, then <i>A</i> can be expressed as<sup id="cite_ref-horn1985_1-0" class="reference"><a href="#cite_note-horn1985-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Golub1996_2-0" class="reference"><a href="#cite_note-Golub1996-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=QUQ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>U</mi> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=QUQ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a55826756ff6249bb9f8c5dd04389d51c42eb45f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.634ex; height:3.009ex;" alt="{\displaystyle A=QUQ^{-1}}"></span> for some <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary matrix</a> <i>Q</i> (so that the inverse <i>Q</i><sup>−1</sup> is also the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> <i>Q</i>* of <i>Q</i>), and some <a href="/wiki/Upper_triangular_matrix" class="mw-redirect" title="Upper triangular matrix">upper triangular matrix</a> <i>U</i>. This is called a <b>Schur form</b> of <i>A</i>. Since <i>U</i> is <a href="/wiki/Similar_(linear_algebra)" class="mw-redirect" title="Similar (linear algebra)">similar</a> to <i>A</i>, it has the same <a href="/wiki/Spectrum_of_a_matrix" title="Spectrum of a matrix">spectrum</a>, and since it is triangular, its <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> are the diagonal entries of <i>U</i>. </p><p>The Schur decomposition implies that there exists a nested sequence of <i>A</i>-invariant subspaces <span class="texhtml">{0} = <i>V</i><sub>0</sub> ⊂ <i>V</i><sub>1</sub> ⊂ ⋯ ⊂ <i>V<sub>n</sub></i> = <b>C</b><sup><i>n</i></sup></span>, and that there exists an ordered <a href="/wiki/Orthonormal_basis" title="Orthonormal basis">orthonormal basis</a> (for the standard <a href="/wiki/Hermitian_form" class="mw-redirect" title="Hermitian form">Hermitian form</a> of <span class="texhtml"><b>C</b><sup><i>n</i></sup></span>) such that the first <i>i</i> basis vectors span <span class="texhtml"><i>V</i><sub><i>i</i></sub></span> for each <i>i</i> occurring in the nested sequence. Phrased somewhat differently, the first part says that a <a href="/wiki/Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> <i>J</i> on a complex finite-dimensional vector space <a href="/wiki/Orbit-stabilizer_theorem#Orbits_and_stabilizers" class="mw-redirect" title="Orbit-stabilizer theorem">stabilizes</a> a complete <a href="/wiki/Flag_(linear_algebra)" title="Flag (linear algebra)">flag</a> <span class="texhtml">(<i>V</i><sub>1</sub>, ..., <i>V<sub>n</sub></i>)</span>. </p><p>There is also a real Schur decomposition. If <span class="texhtml mvar" style="font-style:italic;">A</span> is an <span class="texhtml"><i>n</i> × <i>n</i></span> <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> with <a href="/wiki/Real_numbers" class="mw-redirect" title="Real numbers">real</a> entries, then <i>A</i> can be expressed as<sup id="cite_ref-horn1985-2_4-0" class="reference"><a href="#cite_note-horn1985-2-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=QHQ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>H</mi> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=QHQ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cf735b90bc6b517ba869e2162e0388f0030670e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.915ex; height:3.009ex;" alt="{\displaystyle A=QHQ^{-1}}"></span> where <span class="texhtml mvar" style="font-style:italic;">Q</span> is an <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrix</a> and <span class="texhtml mvar" style="font-style:italic;">H</span> is either upper or lower quasi-triangular. A quasi-triangular matrix is a matrix that when expressed as a block matrix of <span class="texhtml"><i>2</i> × <i>2</i></span> and <span class="texhtml"><i>1</i> × <i>1</i></span> blocks is triangular. This is a stronger property than being <a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a> . Just as in the complex case, a family of commuting real matrices {<i>A<sub>i</sub></i>} may be simultaneously brought to quasi-triangular form by an orthogonal matrix. There exists an orthogonal matrix <i>Q</i> such that, for every <i>A<sub>i</sub></i> in the given family, <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H_{i}=QA_{i}Q^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>Q</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H_{i}=QA_{i}Q^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f096c391c442d0baf0c6f1dbdfc3eff7b130bb5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.382ex; height:3.009ex;" alt="{\displaystyle H_{i}=QA_{i}Q^{-1}}"></span> is upper quasi-triangular. </p> <div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schur_decomposition&action=edit&section=2" title="Edit section: Proof"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A constructive proof for the Schur decomposition is as follows: every operator <i>A</i> on a complex finite-dimensional vector space has an eigenvalue <i>λ</i>, corresponding to some eigenspace <i>V<sub>λ</sub></i>. Let <i>V<sub>λ</sub></i><sup>⊥</sup> be its orthogonal complement. It is clear that, with respect to this orthogonal decomposition, <i>A</i> has matrix representation (one can pick here any orthonormal bases <i>Z</i><sub>1</sub> and <i>Z</i><sub>2</sub> spanning <i>V<sub>λ</sub></i> and <i>V<sub>λ</sub></i><sup>⊥</sup> respectively) <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}^{*}A{\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}={\begin{bmatrix}\lambda \,I_{\lambda }&A_{12}\\0&A_{22}\end{bmatrix}}:{\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}\rightarrow {\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>λ<!-- λ --></mi> <mspace width="thinmathspace" /> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ<!-- λ --></mi> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ<!-- λ --></mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⊕<!-- ⊕ --></mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ<!-- λ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥<!-- ⊥ --></mo> </mrow> </msubsup> </mtd> </mtr> </mtable> </mrow> <mo stretchy="false">→<!-- → --></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ<!-- λ --></mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⊕<!-- ⊕ --></mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ<!-- λ --></mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>⊥<!-- ⊥ --></mo> </mrow> </msubsup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}^{*}A{\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}={\begin{bmatrix}\lambda \,I_{\lambda }&A_{12}\\0&A_{22}\end{bmatrix}}:{\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}\rightarrow {\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5419b37e1304ea03b63d38751dd738e4ff744df" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:52.212ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}^{*}A{\begin{bmatrix}Z_{1}&Z_{2}\end{bmatrix}}={\begin{bmatrix}\lambda \,I_{\lambda }&A_{12}\\0&A_{22}\end{bmatrix}}:{\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}\rightarrow {\begin{matrix}V_{\lambda }\\\oplus \\V_{\lambda }^{\perp }\end{matrix}}}"></span> where <i>I<sub>λ</sub></i> is the identity operator on <i>V<sub>λ</sub></i>. The above matrix would be upper-triangular except for the <i>A</i><sub>22</sub> block. But exactly the same procedure can be applied to the sub-matrix <i>A</i><sub>22</sub>, viewed as an operator on <i>V<sub>λ</sub></i><sup>⊥</sup>, and its submatrices. Continue this way until the resulting matrix is upper triangular. Since each conjugation increases the dimension of the upper-triangular block by at least one, this process takes at most <i>n</i> steps. Thus the space <b>C</b><sup><i>n</i></sup> will be exhausted and the procedure has yielded the desired result.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The above argument can be slightly restated as follows: let <i>λ</i> be an eigenvalue of <i>A</i>, corresponding to some eigenspace <i>V<sub>λ</sub></i>. <i>A</i> induces an operator <i>T</i> on the <a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a> <b>C</b><sup><i>n</i></sup>/<i>V<sub>λ</sub></i>. This operator is precisely the <i>A</i><sub>22</sub> submatrix from above. As before, <i>T</i> would have an eigenspace, say <i>W<sub>μ</sub></i> ⊂ <b>C</b><sup><i>n</i></sup> modulo <i>V<sub>λ</sub></i>. Notice the preimage of <i>W<sub>μ</sub></i> under the quotient map is an <a href="/wiki/Invariant_subspace" title="Invariant subspace">invariant subspace</a> of <i>A</i> that contains <i>V<sub>λ</sub></i>. Continue this way until the resulting quotient space has dimension 0. Then the successive preimages of the eigenspaces found at each step form a flag that <i>A</i> stabilizes. </p> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schur_decomposition&action=edit&section=3" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Although every square matrix has a Schur decomposition, in general this decomposition is not unique. For example, the eigenspace <i>V<sub>λ</sub></i> can have dimension > 1, in which case any orthonormal basis for <i>V<sub>λ</sub></i> would lead to the desired result. </p><p>Write the triangular matrix <i>U</i> as <i>U</i> = <i>D</i> + <i>N</i>, where <i>D</i> is diagonal and <i>N</i> is strictly upper triangular (and thus a <a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">nilpotent matrix</a>). The diagonal matrix <i>D</i> contains the eigenvalues of <i>A</i> in arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of <i>A</i>, while the Frobenius norm of <i>A</i>, squared, is the sum of the squared <a href="/wiki/Singular_value" title="Singular value">singular values</a> of <i>A</i>). The nilpotent part <i>N</i> is generally not unique either, but its <a href="/wiki/Matrix_norm#Frobenius_norm" title="Matrix norm">Frobenius norm</a> is uniquely determined by <i>A</i> (just because the Frobenius norm of A is equal to the Frobenius norm of <i>U</i> = <i>D</i> + <i>N</i>).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p><p>It is clear that if <i>A</i> is a <a href="/wiki/Normal_matrix" title="Normal matrix">normal matrix</a>, then <i>U</i> from its Schur decomposition must be a <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> and the column vectors of <i>Q</i> are the <a href="/wiki/Eigenvector" class="mw-redirect" title="Eigenvector">eigenvectors</a> of <i>A</i>. Therefore, the Schur decomposition extends the <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">spectral decomposition</a>. In particular, if <i>A</i> is <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a>, the Schur decomposition of <i>A</i>, its spectral decomposition, and its <a href="/wiki/Singular_value_decomposition" title="Singular value decomposition">singular value decomposition</a> coincide. </p><p>A <a href="/wiki/Commutative_operation" class="mw-redirect" title="Commutative operation">commuting</a> family {<i>A<sub>i</sub></i>} of matrices can be simultaneously triangularized, i.e. there exists a unitary matrix <i>Q</i> such that, for every <i>A<sub>i</sub></i> in the given family, <i>Q A<sub>i</sub> Q*</i> is upper triangular. This can be readily deduced from the above proof. Take element <i>A</i> from {<i>A<sub>i</sub></i>} and again consider an eigenspace <i>V<sub>A</sub></i>. Then <i>V<sub>A</sub></i> is invariant under all matrices in {<i>A<sub>i</sub></i>}. Therefore, all matrices in {<i>A<sub>i</sub></i>} must share one common eigenvector in <i>V<sub>A</sub></i>. Induction then proves the claim. As a corollary, we have that every commuting family of normal matrices can be simultaneously <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalized</a>. </p><p>In the infinite dimensional setting, not every <a href="/wiki/Bounded_operator" title="Bounded operator">bounded operator</a> on a <a href="/wiki/Banach_space" title="Banach space">Banach space</a> has an invariant subspace. However, the upper-triangularization of an arbitrary square matrix does generalize to <a href="/wiki/Compact_operator" title="Compact operator">compact operators</a>. Every <a href="/wiki/Compact_operator" title="Compact operator">compact operator</a> on a complex Banach space has a <a href="/wiki/Flag_(linear_algebra)#Subspace_nest" title="Flag (linear algebra)">nest</a> of closed invariant subspaces. </p> <div class="mw-heading mw-heading2"><h2 id="Computation">Computation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schur_decomposition&action=edit&section=4" title="Edit section: Computation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Schur decomposition of a given matrix is numerically computed by the <a href="/wiki/QR_algorithm" title="QR algorithm">QR algorithm</a> or its variants. In other words, the roots of the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> corresponding to the matrix are not necessarily computed ahead in order to obtain its Schur decomposition. Conversely, the <a href="/wiki/QR_algorithm" title="QR algorithm">QR algorithm</a> can be used to compute the roots of any given <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> by finding the Schur decomposition of its <a href="/wiki/Companion_matrix" title="Companion matrix">companion matrix</a>. Similarly, the <a href="/wiki/QR_algorithm" title="QR algorithm">QR algorithm</a> is used to compute the eigenvalues of any given matrix, which are the diagonal entries of the upper triangular matrix of the Schur decomposition. Although the <a href="/wiki/QR_algorithm" title="QR algorithm">QR algorithm</a> is formally an infinite sequence of operations, convergence to machine precision is practically achieved in <a href="/wiki/Big_O_notation" title="Big O notation"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}(n^{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}(n^{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff78e74de3bf7a5246829c66bc5acf0c2a94b67c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.108ex; height:3.176ex;" alt="{\displaystyle {\mathcal {O}}(n^{3})}"></span></a> operations.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> See the Nonsymmetric Eigenproblems section in <a href="/wiki/LAPACK" title="LAPACK">LAPACK</a> Users' Guide.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schur_decomposition&action=edit&section=5" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Lie_theory" title="Lie theory">Lie theory</a> applications include: </p> <ul><li>Every invertible operator is contained in a <a href="/wiki/Borel_group" class="mw-redirect" title="Borel group">Borel group</a>.</li> <li>Every operator fixes a point of the <a href="/wiki/Flag_manifold" class="mw-redirect" title="Flag manifold">flag manifold</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Generalized_Schur_decomposition">Generalized Schur decomposition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schur_decomposition&action=edit&section=6" title="Edit section: Generalized Schur decomposition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Given square matrices <i>A</i> and <i>B</i>, the <b>generalized Schur decomposition</b> factorizes both matrices as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=QSZ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi>S</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=QSZ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b844963c492eda3024403ab9406947a351b6857f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.942ex; height:2.676ex;" alt="{\displaystyle A=QSZ^{*}}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B=QTZ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mi>Q</mi> <mi>T</mi> <msup> <mi>Z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B=QTZ^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecc8a60550df19cd526e065cd77a471e4ddd5c21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.1ex; height:2.676ex;" alt="{\displaystyle B=QTZ^{*}}"></span>, where <i>Q</i> and <i>Z</i> are <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a>, and <i>S</i> and <i>T</i> are <a href="/wiki/Upper_triangular" class="mw-redirect" title="Upper triangular">upper triangular</a>. The generalized Schur decomposition is also sometimes called the <b>QZ decomposition</b>.<sup id="cite_ref-Golub1996_2-1" class="reference"><a href="#cite_note-Golub1996-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page: 375">: 375 </span></sup> <sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>The generalized <a href="/wiki/Eigenvalue" class="mw-redirect" title="Eigenvalue">eigenvalues</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ<!-- λ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"></span> that solve the <a href="/wiki/Eigendecomposition_of_a_matrix#Additional_topics" title="Eigendecomposition of a matrix">generalized eigenvalue problem</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} =\lambda B\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mi>λ<!-- λ --></mi> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} =\lambda B\mathbf {x} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0146c75491c67107e4f92e877e235837e3d11336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.783ex; height:2.176ex;" alt="{\displaystyle A\mathbf {x} =\lambda B\mathbf {x} }"></span> (where <b>x</b> is an unknown nonzero vector) can be calculated as the ratio of the diagonal elements of <i>S</i> to those of <i>T</i>. That is, using subscripts to denote matrix elements, the <i>i</i>th generalized eigenvalue <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"></span> satisfies <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}=S_{ii}/T_{ii}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}=S_{ii}/T_{ii}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9809604f643fcd96f98523bbb1c42c925d3dac5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.932ex; height:2.843ex;" alt="{\displaystyle \lambda _{i}=S_{ii}/T_{ii}}"></span>. </p> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Schur_decomposition&action=edit&section=7" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-horn1985-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-horn1985_1-0">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFHornJohnson1985" class="citation book cs1">Horn, R.A. & Johnson, C.R. (1985). <i>Matrix Analysis</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-38632-2" title="Special:BookSources/0-521-38632-2"><bdi>0-521-38632-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis&rft.pub=Cambridge+University+Press&rft.date=1985&rft.isbn=0-521-38632-2&rft.aulast=Horn&rft.aufirst=R.A.&rft.au=Johnson%2C+C.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span> (Section 2.3 and further at <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PlYQN0ypTwEC&pg=PA79&dq=%22Schur%22">p. 79</a>)</span> </li> <li id="cite_note-Golub1996-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Golub1996_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Golub1996_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubVan_Loan1996" class="citation book cs1">Golub, G.H. & Van Loan, C.F. (1996). <i>Matrix Computations</i> (3rd ed.). Johns Hopkins University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-8018-5414-8" title="Special:BookSources/0-8018-5414-8"><bdi>0-8018-5414-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Computations&rft.edition=3rd&rft.pub=Johns+Hopkins+University+Press&rft.date=1996&rft.isbn=0-8018-5414-8&rft.aulast=Golub&rft.aufirst=G.H.&rft.au=Van+Loan%2C+C.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span>(Section 7.7 at <a rel="nofollow" class="external text" href="https://books.google.com/books?id=mlOa7wPX6OYC&pg=PA313&dq=%22Schur+Decomposition%22">p. 313</a>)</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchott2016" class="citation book cs1">Schott, James R. (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=e-JFDAAAQBAJ&pg=PA177"><i>Matrix Analysis for Statistics</i></a> (3rd ed.). New York: John Wiley & Sons. pp. <span class="nowrap">175–</span>178. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-119-09247-6" title="Special:BookSources/978-1-119-09247-6"><bdi>978-1-119-09247-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis+for+Statistics&rft.place=New+York&rft.pages=%3Cspan+class%3D%22nowrap%22%3E175-%3C%2Fspan%3E178&rft.edition=3rd&rft.pub=John+Wiley+%26+Sons&rft.date=2016&rft.isbn=978-1-119-09247-6&rft.aulast=Schott&rft.aufirst=James+R.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3De-JFDAAAQBAJ%26pg%3DPA177&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span></span> </li> <li id="cite_note-horn1985-2-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-horn1985-2_4-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHornJohnson1985" class="citation book cs1">Horn, R.A. & Johnson, C.R. (1985). <i>Matrix Analysis</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-38632-2" title="Special:BookSources/0-521-38632-2"><bdi>0-521-38632-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Analysis&rft.pub=Cambridge+University+Press&rft.date=1985&rft.isbn=0-521-38632-2&rft.aulast=Horn&rft.aufirst=R.A.&rft.au=Johnson%2C+C.R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span> (Section 2.3 and further at <a rel="nofollow" class="external text" href="https://books.google.com/books?id=PlYQN0ypTwEC&pg=PA82&dq=%222.3.4%22">p. 82</a>)</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWagner" class="citation web cs1">Wagner, David. <a rel="nofollow" class="external text" href="https://math.mit.edu/~gs/linearalgebra/ila5/lafe_schur03.pdf">"Proof of Schur's Theorem"</a> <span class="cs1-format">(PDF)</span>. <i>Notes on Linear Algebra</i>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Notes+on+Linear+Algebra&rft.atitle=Proof+of+Schur%27s+Theorem&rft.aulast=Wagner&rft.aufirst=David&rft_id=https%3A%2F%2Fmath.mit.edu%2F~gs%2Flinearalgebra%2Fila5%2Flafe_schur03.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHigham2022" class="citation web cs1">Higham, Nick (11 May 2022). <a rel="nofollow" class="external text" href="https://nhigham.com/2022/05/11/what-is-a-schur-decomposition/">"What Is a Schur Decomposition?"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=What+Is+a+Schur+Decomposition%3F&rft.date=2022-05-11&rft.aulast=Higham&rft.aufirst=Nick&rft_id=https%3A%2F%2Fnhigham.com%2F2022%2F05%2F11%2Fwhat-is-a-schur-decomposition%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTrefethenBau1997" class="citation book cs1">Trefethen, Lloyd N.; Bau, David (1997). <a rel="nofollow" class="external text" href="https://www.worldcat.org/oclc/36084666"><i>Numerical linear algebra</i></a>. Philadelphia: Society for Industrial and Applied Mathematics. pp. <span class="nowrap">193–</span>194. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-361-7" title="Special:BookSources/0-89871-361-7"><bdi>0-89871-361-7</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36084666">36084666</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+linear+algebra&rft.place=Philadelphia&rft.pages=%3Cspan+class%3D%22nowrap%22%3E193-%3C%2Fspan%3E194&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1997&rft_id=info%3Aoclcnum%2F36084666&rft.isbn=0-89871-361-7&rft.aulast=Trefethen&rft.aufirst=Lloyd+N.&rft.au=Bau%2C+David&rft_id=https%3A%2F%2Fwww.worldcat.org%2Foclc%2F36084666&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndersonBaiBischofBlackford1995" class="citation book cs1">Anderson, E; Bai, Z; Bischof, C; Blackford, S; Demmel, J; Dongarra, J; Du Croz, J; Greenbaum, A; Hammarling, S; McKenny, A; Sorensen, D (1995). <a rel="nofollow" class="external text" href="http://www.netlib.org/lapack/lug/"><i>LAPACK Users guide</i></a>. Philadelphia, PA: Society for Industrial and Applied Mathematics. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-447-8" title="Special:BookSources/0-89871-447-8"><bdi>0-89871-447-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=LAPACK+Users+guide&rft.place=Philadelphia%2C+PA&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1995&rft.isbn=0-89871-447-8&rft.aulast=Anderson&rft.aufirst=E&rft.au=Bai%2C+Z&rft.au=Bischof%2C+C&rft.au=Blackford%2C+S&rft.au=Demmel%2C+J&rft.au=Dongarra%2C+J&rft.au=Du+Croz%2C+J&rft.au=Greenbaum%2C+A&rft.au=Hammarling%2C+S&rft.au=McKenny%2C+A&rft.au=Sorensen%2C+D&rft_id=http%3A%2F%2Fwww.netlib.org%2Flapack%2Flug%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASchur+decomposition" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Daniel Kressner: "Numerical Methods for General and Structured Eigenvalue Problems", Chap-2, Springer, LNCSE-46 (2005).</span> </li> </ol></div> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐65b64b4b74‐j7w22 Cached time: 20250219124836 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.245 seconds Real time usage: 0.371 seconds Preprocessor visited node count: 1403/1000000 Post‐expand include size: 16383/2097152 bytes Template argument size: 1363/2097152 bytes Highest expansion depth: 15/100 Expensive parser function count: 1/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 28880/5000000 bytes Lua time usage: 0.136/10.000 seconds Lua memory usage: 5102562/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 250.170 1 -total 48.82% 122.131 6 Template:Cite_book 25.13% 62.857 1 Template:Short_description 16.14% 40.367 2 Template:Pagetype 8.89% 22.249 1 Template:Rp 7.03% 17.575 1 Template:R/superscript 5.48% 13.697 2 Template:Cite_web 5.17% 12.922 10 Template:Main_other 4.61% 11.537 8 Template:Math 3.93% 9.825 1 Template:SDcat --> <!-- Saved in parser cache with key enwiki:pcache:372514:|#|:idhash:canonical and timestamp 20250219124836 and revision id 1269572295. Rendering was triggered because: page-view --> </div><!--esi <esi:include src="/esitest-fa8a495983347898/content" /> --><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Schur_decomposition&oldid=1269572295">https://en.wikipedia.org/w/index.php?title=Schur_decomposition&oldid=1269572295</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Matrix_theory" title="Category:Matrix theory">Matrix theory</a></li><li><a href="/wiki/Category:Matrix_decompositions" title="Category:Matrix decompositions">Matrix decompositions</a></li><li><a href="/wiki/Category:Issai_Schur" title="Category:Issai Schur">Issai Schur</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li><li><a href="/wiki/Category:Articles_containing_proofs" title="Category:Articles containing proofs">Articles containing proofs</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 15 January 2025, at 09:26<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Schur_decomposition&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/static/images/footer/wikimedia-button.svg" width="84" height="29"><img src="/static/images/footer/wikimedia.svg" width="25" height="25" alt="Wikimedia Foundation" lang="en" loading="lazy"></picture></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><picture><source media="(min-width: 500px)" srcset="/w/resources/assets/poweredby_mediawiki.svg" width="88" height="31"><img src="/w/resources/assets/mediawiki_compact.svg" alt="Powered by MediaWiki" width="25" height="25" loading="lazy"></picture></a></li> </ul> </footer> </div> </div> </div> <div class="vector-header-container vector-sticky-header-container"> <div id="vector-sticky-header" class="vector-sticky-header"> <div class="vector-sticky-header-start"> <div class="vector-sticky-header-icon-start vector-button-flush-left vector-button-flush-right" aria-hidden="true"> <button class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-sticky-header-search-toggle" tabindex="-1" data-event-name="ui.vector-sticky-search-form.icon"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </button> </div> <div role="search" class="vector-search-box-vue vector-search-box-show-thumbnail vector-search-box"> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail"> <form action="/w/index.php" id="vector-sticky-search-form" class="cdx-search-input cdx-search-input--has-end-button"> <div class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia"> <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <div class="vector-sticky-header-context-bar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-sticky-header-toc" class="vector-dropdown mw-portlet mw-portlet-sticky-header-toc vector-sticky-header-toc vector-button-flush-left" > <input type="checkbox" id="vector-sticky-header-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-sticky-header-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-sticky-header-toc-label" for="vector-sticky-header-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-sticky-header-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">Schur decomposition</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-history-sticky-header" tabindex="-1" data-event-name="history-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-history mw-ui-icon-wikimedia-wikimedia-history"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only mw-watchlink" id="ca-watchstar-sticky-header" tabindex="-1" data-event-name="watch-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-star mw-ui-icon-wikimedia-wikimedia-star"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-edit-sticky-header" tabindex="-1" data-event-name="wikitext-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-wikiText mw-ui-icon-wikimedia-wikimedia-wikiText"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-ve-edit-sticky-header" tabindex="-1" data-event-name="ve-edit-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-edit mw-ui-icon-wikimedia-wikimedia-edit"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-viewsource-sticky-header" tabindex="-1" data-event-name="ve-edit-protected-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-editLock mw-ui-icon-wikimedia-wikimedia-editLock"></span> <span></span> </a> </div> <div class="vector-sticky-header-buttons"> <button class="cdx-button cdx-button--weight-quiet mw-interlanguage-selector" id="p-lang-btn-sticky-header" tabindex="-1" data-event-name="ui.dropdown-p-lang-btn-sticky-header"><span class="vector-icon mw-ui-icon-wikimedia-language mw-ui-icon-wikimedia-wikimedia-language"></span> <span>14 languages</span> </button> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.canary-5744cb454d-ftrkb","wgBackendResponseTime":139,"wgPageParseReport":{"limitreport":{"cputime":"0.245","walltime":"0.371","ppvisitednodes":{"value":1403,"limit":1000000},"postexpandincludesize":{"value":16383,"limit":2097152},"templateargumentsize":{"value":1363,"limit":2097152},"expansiondepth":{"value":15,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":28880,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 250.170 1 -total"," 48.82% 122.131 6 Template:Cite_book"," 25.13% 62.857 1 Template:Short_description"," 16.14% 40.367 2 Template:Pagetype"," 8.89% 22.249 1 Template:Rp"," 7.03% 17.575 1 Template:R/superscript"," 5.48% 13.697 2 Template:Cite_web"," 5.17% 12.922 10 Template:Main_other"," 4.61% 11.537 8 Template:Math"," 3.93% 9.825 1 Template:SDcat"]},"scribunto":{"limitreport-timeusage":{"value":"0.136","limit":"10.000"},"limitreport-memusage":{"value":5102562,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-65b64b4b74-j7w22","timestamp":"20250219124836","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Schur decomposition","url":"https:\/\/en.wikipedia.org\/wiki\/Schur_decomposition","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1064218","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1064218","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2003-11-19T19:17:03Z","dateModified":"2025-01-15T09:26:52Z","headline":"matrix decomposition"}</script> </body> </html>