矩阵指数 - 维基百科，自由的百科全书

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-toc-available" lang="zh" dir="ltr"> <head> <meta charset="UTF-8"> <title>矩阵指数 - 维基百科，自由的百科全书</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-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-toc-available";var cookie=document.cookie.match(/(?:^|; )zhwikimwclientpreferences=([^;]+)/);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":"zh", "wgMonthNames":["","1月","2月","3月","4月","5月","6月","7月","8月","9月","10月","11月","12月"],"wgRequestId":"b4735490-2d0b-45a8-a754-5a169d8a99ed","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"矩阵指数","wgTitle":"矩阵指数","wgCurRevisionId":84499588,"wgRevisionId":84499588,"wgArticleId":824877,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["含有哈佛参考文献格式系列模板链接指向错误的页面","矩陣論","李群","指数"],"wgPageViewLanguage":"zh","wgPageContentLanguage":"zh","wgPageContentModel":"wikitext","wgRelevantPageName":"矩阵指数","wgRelevantArticleId":824877,"wgUserVariant":"zh","wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[],"wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":true,"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault": true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"zh","pageLanguageDir":"ltr","pageVariantFallbacks":["zh-hans","zh-hant","zh-cn","zh-tw","zh-hk","zh-sg","zh-mo","zh-my"]},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":true,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q1191722","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false,"wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.gadget.large-font":"ready","ext.globalCssJs.user.styles":"ready","site.styles":"ready", "user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.math.styles":"ready","ext.cite.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=["ext.cite.ux-enhancements","ext.scribunto.logs","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.edit0","ext.gadget.WikiMiniAtlas","ext.gadget.UnihanTooltips","ext.gadget.Difflink","ext.gadget.pseudonamespace-UI","ext.gadget.SpecialWikitext","ext.gadget.switcher","ext.gadget.VariantAlly","ext.gadget.AdvancedSiteNotices","ext.gadget.hideConversionTab","ext.gadget.internalLinkHelper-altcolor","ext.gadget.noteTA","ext.gadget.NavFrame","ext.gadget.collapsibleTables", "ext.gadget.scrollUpButton","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.quicksurveys.init","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</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=zh&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=zh&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=ext.gadget.large-font&only=styles&skin=vector-2022"> <link rel="stylesheet" href="/w/load.php?lang=zh&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.5"> <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="矩阵指数 - 维基百科，自由的百科全书"> <meta property="og:type" content="website"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//zh.m.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" type="application/x-wiki" title="编辑本页" href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&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 (zh)"> <link rel="EditURI" type="application/rsd+xml" href="//zh.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://zh.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh" href="https://zh.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hans" href="https://zh.wikipedia.org/zh-hans/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hans-CN" href="https://zh.wikipedia.org/zh-cn/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hans-MY" href="https://zh.wikipedia.org/zh-my/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hans-SG" href="https://zh.wikipedia.org/zh-sg/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hant" href="https://zh.wikipedia.org/zh-hant/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hant-HK" href="https://zh.wikipedia.org/zh-hk/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hant-MO" href="https://zh.wikipedia.org/zh-mo/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="zh-Hant-TW" href="https://zh.wikipedia.org/zh-tw/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="alternate" hreflang="x-default" href="https://zh.wikipedia.org/wiki/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.zh"> <link rel="alternate" type="application/atom+xml" title="Wikipedia的Atom feed" href="/w/index.php?title=Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9&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-矩阵指数 rootpage-矩阵指数 skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">跳转到内容</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="站点"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <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="主菜单" > <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">主菜单</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">主菜单</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">隐藏</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> 导航 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" title="访问首页[z]" accesskey="z"><span>首页</span></a></li><li id="n-indexpage" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%88%86%E7%B1%BB%E7%B4%A2%E5%BC%95" title="以分类索引搜寻中文维基百科"><span>分类索引</span></a></li><li id="n-Featured_content" class="mw-list-item"><a href="/wiki/Portal:%E7%89%B9%E8%89%B2%E5%85%A7%E5%AE%B9" title="查看中文维基百科的特色内容"><span>特色内容</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:%E6%96%B0%E8%81%9E%E5%8B%95%E6%85%8B" title="提供当前新闻事件的背景资料"><span>新闻动态</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:%E6%9C%80%E8%BF%91%E6%9B%B4%E6%94%B9" title="列出维基百科中的最近修改[r]" accesskey="r"><span>最近更改</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:%E9%9A%8F%E6%9C%BA%E9%A1%B5%E9%9D%A2" title="随机载入一个页面[x]" accesskey="x"><span>随机条目</span></a></li> </ul> </div> </div> <div id="p-help" class="vector-menu mw-portlet mw-portlet-help" > <div class="vector-menu-heading"> 帮助 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:%E7%9B%AE%E5%BD%95" title="寻求帮助"><span>帮助</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%A4%BE%E7%BE%A4%E9%A6%96%E9%A1%B5" title="关于本计划、你可以做什么、应该如何做"><span>维基社群</span></a></li><li id="n-policy" class="mw-list-item"><a href="/wiki/Wikipedia:%E6%96%B9%E9%87%9D%E8%88%87%E6%8C%87%E5%BC%95" title="查看维基百科的方针和指引"><span>方针与指引</span></a></li><li id="n-villagepump" class="mw-list-item"><a href="/wiki/Wikipedia:%E4%BA%92%E5%8A%A9%E5%AE%A2%E6%A0%88" title="参与维基百科社群的讨论"><span>互助客栈</span></a></li><li id="n-Information_desk" class="mw-list-item"><a href="/wiki/Wikipedia:%E7%9F%A5%E8%AF%86%E9%97%AE%E7%AD%94" title="解答任何与维基百科无关的问题的地方"><span>知识问答</span></a></li><li id="n-conversion" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%AD%97%E8%AF%8D%E8%BD%AC%E6%8D%A2" title="提出字词转换请求"><span>字词转换</span></a></li><li id="n-IRC" class="mw-list-item"><a href="/wiki/Wikipedia:IRC%E8%81%8A%E5%A4%A9%E9%A2%91%E9%81%93"><span>IRC即时聊天</span></a></li><li id="n-contact" class="mw-list-item"><a href="/wiki/Wikipedia:%E8%81%94%E7%BB%9C%E6%88%91%E4%BB%AC" title="如何联络维基百科"><span>联络我们</span></a></li><li id="n-about" class="mw-list-item"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E" title="查看维基百科的简介"><span>关于维基百科</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Wikipedia:%E9%A6%96%E9%A1%B5" 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="维基百科" src="/static/images/mobile/copyright/wikipedia-wordmark-zh.svg" style="width: 6.5625em; height: 1.375em;"> <img class="mw-logo-tagline" alt="自由的百科全书" src="/static/images/mobile/copyright/wikipedia-tagline-zh.svg" width="103" height="14" style="width: 6.4375em; height: 0.875em;"> </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:%E6%90%9C%E7%B4%A2" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="搜索维基百科[f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>搜索</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="搜索维基百科" aria-label="搜索维基百科" autocapitalize="sentences" title="搜索维基百科[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:搜索"> </div> <button class="cdx-button cdx-search-input__end-button">搜索</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="个人工具"> <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="外观"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="更改页面字体大小、宽度和颜色的外观" > <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="外观" > <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">外观</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=zh.wikipedia.org&uselang=zh" class=""><span>资助维基百科</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:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" title="我们推荐您创建账号并登录，但这不是强制性的" class=""><span>创建账号</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:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" title="建议你登录，尽管并非必须。[o]" accesskey="o" class=""><span>登录</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="更多选项" > <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="个人工具" > <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">个人工具</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="用户菜单" > <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=zh.wikipedia.org&uselang=zh"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</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"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</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:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y"><span>贡献</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n"><span>讨论</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"></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="站点"> <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="目录" 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">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</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">序言</div> </a> </li> <li id="toc-性质" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#性质"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>性质</span> </div> </a> <button aria-controls="toc-性质-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关性质子章节</span> </button> <ul id="toc-性质-sublist" class="vector-toc-list"> <li id="toc-基本性质" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#基本性质"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>基本性质</span> </div> </a> <ul id="toc-基本性质-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-线性微分方程" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#线性微分方程"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>线性微分方程</span> </div> </a> <ul id="toc-线性微分方程-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-矩阵指数的行列式" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#矩阵指数的行列式"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>矩阵指数的行列式</span> </div> </a> <ul id="toc-矩阵指数的行列式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-指数相加" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#指数相加"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.4</span> <span>指数相加</span> </div> </a> <ul id="toc-指数相加-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-李乘积公式" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#李乘积公式"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.5</span> <span>李乘积公式</span> </div> </a> <ul id="toc-李乘积公式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-贝克尔-坎贝尔-豪斯多夫公式" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#贝克尔-坎贝尔-豪斯多夫公式"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.6</span> <span>贝克尔-坎贝尔-豪斯多夫公式</span> </div> </a> <ul id="toc-贝克尔-坎贝尔-豪斯多夫公式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-指数映射" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#指数映射"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.7</span> <span>指数映射</span> </div> </a> <ul id="toc-指数映射-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-矩阵指数的计算" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#矩阵指数的计算"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>矩阵指数的计算</span> </div> </a> <button aria-controls="toc-矩阵指数的计算-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关矩阵指数的计算子章节</span> </button> <ul id="toc-矩阵指数的计算-sublist" class="vector-toc-list"> <li id="toc-可对角化矩阵" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#可对角化矩阵"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>可对角化矩阵</span> </div> </a> <ul id="toc-可对角化矩阵-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-幂零矩阵" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#幂零矩阵"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>幂零矩阵</span> </div> </a> <ul id="toc-幂零矩阵-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-推广" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#推广"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>推广</span> </div> </a> <ul id="toc-推广-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-计算" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#计算"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>计算</span> </div> </a> <ul id="toc-计算-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-应用" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#应用"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>应用</span> </div> </a> <button aria-controls="toc-应用-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关应用子章节</span> </button> <ul id="toc-应用-sublist" class="vector-toc-list"> <li id="toc-线性微分方程_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#线性微分方程_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>线性微分方程</span> </div> </a> <ul id="toc-线性微分方程_2-sublist" class="vector-toc-list"> <li id="toc-例子（齐次）" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#例子（齐次）"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.1</span> <span>例子（齐次）</span> </div> </a> <ul id="toc-例子（齐次）-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-非齐次的情况──参数变换" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#非齐次的情况──参数变换"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.2</span> <span>非齐次的情况──参数变换</span> </div> </a> <ul id="toc-非齐次的情况──参数变换-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-例子（非齐次）" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#例子（非齐次）"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1.3</span> <span>例子（非齐次）</span> </div> </a> <ul id="toc-例子（非齐次）-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-註釋" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#註釋"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>註釋</span> </div> </a> <ul id="toc-註釋-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参閱" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参閱"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>参閱</span> </div> </a> <ul id="toc-参閱-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>外部链接</span> </div> </a> <ul id="toc-外部链接-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="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <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="开关目录" > <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">开关目录</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">矩阵指数</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="前往另一种语言写成的文章。19种语言可用" > <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-19" 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">19种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D1%80%D0%B8%D1%87%D0%BD%D0%B0_%D0%B5%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%82%D0%B0" title="Матрична експонента – 保加利亚语" lang="bg" hreflang="bg" data-title="Матрична експонента" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Exponencial_d%27una_matriu" title="Exponencial d'una matriu – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Exponencial d'una matriu" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" 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/Maticov%C3%A1_exponenci%C3%A1la" title="Maticová exponenciála – 捷克语" lang="cs" hreflang="cs" data-title="Maticová exponenciála" data-language-autonym="Čeština" data-language-local-name="捷克语" 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/Matrixexponential" title="Matrixexponential – 德语" lang="de" hreflang="de" data-title="Matrixexponential" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Matrix_exponential" title="Matrix exponential – 英语" lang="en" hreflang="en" data-title="Matrix exponential" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Exponencial_de_una_matriz" title="Exponencial de una matriz – 西班牙语" lang="es" hreflang="es" data-title="Exponencial de una matriz" data-language-autonym="Español" data-language-local-name="西班牙语" 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/Exponentielle_d%27une_matrice" title="Exponentielle d'une matrice – 法语" lang="fr" hreflang="fr" data-title="Exponentielle d'une matrice" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%90%D7%A7%D7%A1%D7%A4%D7%95%D7%A0%D7%A0%D7%98_%D7%A9%D7%9C_%D7%9E%D7%98%D7%A8%D7%99%D7%A6%D7%95%D7%AA" title="אקספוננט של מטריצות – 希伯来语" lang="he" hreflang="he" data-title="אקספוננט של מטריצות" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matrice_esponenziale" title="Matrice esponenziale – 意大利语" lang="it" hreflang="it" data-title="Matrice esponenziale" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E8%A1%8C%E5%88%97%E6%8C%87%E6%95%B0%E9%96%A2%E6%95%B0" title="行列指数関数 – 日语" lang="ja" hreflang="ja" data-title="行列指数関数" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%96%89%EB%A0%AC_%EC%A7%80%EC%88%98_%ED%95%A8%EC%88%98" title="행렬 지수 함수 – 韩语" lang="ko" hreflang="ko" data-title="행렬 지수 함수" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Eksponenta_macierzy" title="Eksponenta macierzy – 波兰语" lang="pl" hreflang="pl" data-title="Eksponenta macierzy" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Exponencial_matricial" title="Exponencial matricial – 葡萄牙语" lang="pt" hreflang="pt" data-title="Exponencial matricial" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%AD%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%82%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8B" title="Экспонента матрицы – 俄语" lang="ru" hreflang="ru" data-title="Экспонента матрицы" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Eksponent_matrike" title="Eksponent matrike – 斯洛文尼亚语" lang="sl" hreflang="sl" data-title="Eksponent matrike" data-language-autonym="Slovenščina" data-language-local-name="斯洛文尼亚语" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Eksponenciali_matricor" title="Eksponenciali matricor – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Eksponenciali matricor" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matrisexponentialfunktion" title="Matrisexponentialfunktion – 瑞典语" lang="sv" hreflang="sv" data-title="Matrisexponentialfunktion" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%95%D0%BA%D1%81%D0%BF%D0%BE%D0%BD%D0%B5%D0%BD%D1%82%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%96" title="Експонента матриці – 乌克兰语" lang="uk" hreflang="uk" data-title="Експонента матриці" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%AA%D9%88%D8%B6%DB%8C%D8%AD%DB%8C_%D9%82%D8%A7%D9%84%D8%A8" title="توضیحی قالب – 乌尔都语" lang="ur" hreflang="ur" data-title="توضیحی قالب" data-language-autonym="اردو" data-language-local-name="乌尔都语" class="interlanguage-link-target"><span>اردو</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/Q1191722#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</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="命名空间"> <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/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" rel="discussion" title="关于此页面的讨论[t]" accesskey="t"><span>讨论</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown " > <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="更改语言变体" > <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">不转换</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-varlang-0" class="selected ca-variants-zh mw-list-item"><a href="/zh/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh" hreflang="zh"><span>不转换</span></a></li><li id="ca-varlang-1" class="ca-variants-zh-Hans mw-list-item"><a href="/zh-hans/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hans" hreflang="zh-Hans"><span>简体</span></a></li><li id="ca-varlang-2" class="ca-variants-zh-Hant mw-list-item"><a href="/zh-hant/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hant" hreflang="zh-Hant"><span>繁體</span></a></li><li id="ca-varlang-3" class="ca-variants-zh-Hans-CN mw-list-item"><a href="/zh-cn/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hans-CN" hreflang="zh-Hans-CN"><span>大陆简体</span></a></li><li id="ca-varlang-4" class="ca-variants-zh-Hant-HK mw-list-item"><a href="/zh-hk/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hant-HK" hreflang="zh-Hant-HK"><span>香港繁體</span></a></li><li id="ca-varlang-5" class="ca-variants-zh-Hant-MO mw-list-item"><a href="/zh-mo/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hant-MO" hreflang="zh-Hant-MO"><span>澳門繁體</span></a></li><li id="ca-varlang-6" class="ca-variants-zh-Hans-MY mw-list-item"><a href="/zh-my/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hans-MY" hreflang="zh-Hans-MY"><span>大马简体</span></a></li><li id="ca-varlang-7" class="ca-variants-zh-Hans-SG mw-list-item"><a href="/zh-sg/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hans-SG" hreflang="zh-Hans-SG"><span>新加坡简体</span></a></li><li id="ca-varlang-8" class="ca-variants-zh-Hant-TW mw-list-item"><a href="/zh-tw/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" lang="zh-Hant-TW" hreflang="zh-Hant-TW"><span>臺灣正體</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="查看"> <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/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"><span>阅读</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=history" title="本页面的早前版本。[h]" accesskey="h"><span>查看历史</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="页面工具"> <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="工具" > <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">工具</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">工具</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">隐藏</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="更多选项" > <div class="vector-menu-heading"> 操作 </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/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0"><span>阅读</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit" title="编辑该页面[e]" accesskey="e"><span>编辑</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=history"><span>查看历史</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> 常规 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%85%A5%E9%A1%B5%E9%9D%A2/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" title="列出所有与本页相链的页面[j]" accesskey="j"><span>链入页面</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:%E9%93%BE%E5%87%BA%E6%9B%B4%E6%94%B9/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0" rel="nofollow" title="页面链出所有页面的更改[k]" accesskey="k"><span>相关更改</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Project:%E4%B8%8A%E4%BC%A0" title="上传图像或多媒体文件[u]" accesskey="u"><span>上传文件</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:%E7%89%B9%E6%AE%8A%E9%A1%B5%E9%9D%A2" title="全部特殊页面的列表[q]" accesskey="q"><span>特殊页面</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&oldid=84499588" title="此页面该修订版本的固定链接"><span>固定链接</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=info" title="关于此页面的更多信息"><span>页面信息</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:%E5%BC%95%E7%94%A8%E6%AD%A4%E9%A1%B5%E9%9D%A2&page=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&id=84499588&wpFormIdentifier=titleform" title="有关如何引用此页面的信息"><span>引用此页</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:URL%E7%BC%A9%E7%9F%AD%E7%A8%8B%E5%BA%8F&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%259F%25A9%25E9%2598%25B5%25E6%258C%2587%25E6%2595%25B0"><span>获取短链接</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fzh.wikipedia.org%2Fwiki%2F%25E7%259F%25A9%25E9%2598%25B5%25E6%258C%2587%25E6%2595%25B0"><span>下载二维码</span></a></li> </ul> </div> </div> <div id="p-electronpdfservice-sidebar-portlet-heading" class="vector-menu mw-portlet mw-portlet-electronpdfservice-sidebar-portlet-heading" > <div class="vector-menu-heading"> 打印/导出 </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="electron-print_pdf" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=show-download-screen"><span>下载为PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="javascript:print();" rel="alternate" title="本页面的可打印版本[p]" accesskey="p"><span>打印页面</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"> 在其他项目中 </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/Q1191722" title="链接到连接的数据仓库项目[g]" accesskey="g"><span>维基数据项目</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="页面工具"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="外观"> <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">外观</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">隐藏</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">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><p><b>矩阵指数</b>（matrix exponential）是<a href="/wiki/%E6%96%B9%E5%9D%97%E7%9F%A9%E9%98%B5" title="方块矩阵">方块矩阵</a>的一种<a href="/wiki/%E7%9F%A9%E9%98%B5%E5%87%BD%E6%95%B0" title="矩阵函数">矩阵函数</a>，与<a href="/wiki/%E6%8C%87%E6%95%B0%E5%87%BD%E6%95%B0" title="指数函数">指数函数</a>类似。矩阵指数给出了矩阵<a href="/wiki/%E6%9D%8E%E4%BB%A3%E6%95%B0" class="mw-redirect" title="李代数">李代数</a>与对应的<a href="/wiki/%E6%9D%8E%E7%BE%A4" title="李群">李群</a>之间的关系。 </p><p>设<i>X</i>为<i>n</i>×<i>n</i>的<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a>或<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a><a href="/wiki/%E7%9F%A9%E9%98%B5" title="矩阵">矩阵</a>。<i>X</i>的指数，用<i>e</i><sup><i>X</i></sup>或exp(<i>X</i>)来表示，是由以下<a href="/wiki/%E5%B9%82%E7%BA%A7%E6%95%B0" title="幂级数">幂级数</a>所给出的<i>n</i>×<i>n</i>矩阵： </p> <dl><dd><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 e^{X}=\sum _{k=0}^{\infty }{1 \over k!}X^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X}=\sum _{k=0}^{\infty }{1 \over k!}X^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a51cff6614a7f85edea3a64e83231d4c4b307cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:15.336ex; height:7.009ex;" alt="{\displaystyle e^{X}=\sum _{k=0}^{\infty }{1 \over k!}X^{k}}"></span></dd></dl> <p>以上的级数总是收敛的，因此<i>X</i>的指数是定义良好的。注意，如果<i>X</i>是1×1的矩阵，则<i>X</i>的矩阵指数就是由<i>X</i>的元素的指数所组成的1×1矩阵。 </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="性质"><span id=".E6.80.A7.E8.B4.A8"></span>性质</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=1" title="编辑章节：性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="基本性质"><span id=".E5.9F.BA.E6.9C.AC.E6.80.A7.E8.B4.A8"></span>基本性质</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=2" title="编辑章节：基本性质"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>设<i>X</i>和<i>Y</i>为<i>n</i>×<i>n</i>的复数矩阵，并设<i>a</i>和<i>b</i>为任意的复数。我们把<i>n</i>×<i>n</i>的<a href="/wiki/%E5%8D%95%E4%BD%8D%E7%9F%A9%E9%98%B5" class="mw-redirect" title="单位矩阵">单位矩阵</a>记为<i>I</i>，把<a href="/wiki/%E9%9B%B6%E7%9F%A9%E9%98%B5" class="mw-redirect" title="零矩阵">零矩阵</a>记为0。 </p><p>我们可以从指数级数的定义直接得到矩阵指数的如下性质<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>： </p> <ul><li><style data-mw-deduplicate="TemplateStyles:r58896141">.mw-parser-output .serif{font-family:Times,serif}</style><span class="serif"><span class="texhtml"><i>e</i><sup>0</sup> = <i>I</i></span></span></li> <li>exp(<i>X</i><sup>T</sup>) = (exp <i>X</i>)<sup>T</sup>，其中<i>X</i><sup>T</sup>表示<i>X</i>的<a href="/wiki/%E8%BD%89%E7%BD%AE" class="mw-redirect" title="轉置">转置</a>。从中可以推出，如果<i>X</i>是<a href="/wiki/%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" title="對稱矩陣">对称矩阵</a>，则<i>e</i><sup><i>X</i></sup>也是对称矩阵；如果<i>X</i>是<a href="/wiki/%E6%96%9C%E5%B0%8D%E7%A8%B1%E7%9F%A9%E9%99%A3" class="mw-redirect" title="斜對稱矩陣">斜对称矩阵</a>，则<i>e</i><sup><i>X</i></sup>是<a href="/wiki/%E6%AD%A3%E4%BA%A4%E7%9F%A9%E9%98%B5" title="正交矩阵">正交矩阵</a>。</li></ul> <ul><li>exp(<i>X</i>*) = (exp <i>X</i>)*，其中<i>X</i>*表示<i>X</i>的<a href="/wiki/%E5%85%B1%E8%BD%AD%E8%BD%AC%E7%BD%AE" title="共轭转置">共轭转置</a>。可以推出，如果<i>X</i>是<a href="/wiki/%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="埃尔米特矩阵">埃尔米特矩阵</a>，则<i>e</i><sup><i>X</i></sup>也是埃尔米特矩阵；如果<i>X</i>是<a href="/wiki/%E6%96%9C%E5%9F%83%E5%B0%94%E7%B1%B3%E7%89%B9%E7%9F%A9%E9%98%B5" title="斜埃尔米特矩阵">斜埃尔米特矩阵</a>，则<i>e</i><sup><i>X</i></sup>是<a href="/wiki/%E9%85%89%E7%9F%A9%E9%98%B5" title="酉矩阵">酉矩阵</a>。</li></ul> <ul><li>如果<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml"><i>Y</i></span></span>是<a href="/wiki/%E5%8F%AF%E9%80%86%E7%9F%A9%E9%98%B5" class="mw-redirect" title="可逆矩阵">可逆矩阵</a>，那么 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml"><i>e</i><sup><i>YXY</i><sup>−1</sup></sup> = <i>Ye</i><sup><i>X</i></sup><i>Y</i><sup>−1</sup> </span></span></li></ul> <p>接下来是一个关键性质： </p> <ul><li>如果<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 XY=YX}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mi>Y</mi> <mo>=</mo> <mi>Y</mi> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle XY=YX}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/543dd8aea552a72267833b6c99ac36e21bb52b2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.605ex; height:2.176ex;" alt="{\displaystyle XY=YX}"></span>那么 <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 e^{X}e^{Y}=e^{X+Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X}e^{Y}=e^{X+Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613aaf6f80251e7b95e48edb2da3fe9f0e408988" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.632ex; height:2.676ex;" alt="{\displaystyle e^{X}e^{Y}=e^{X+Y}}"></span></li></ul> <p>由此导出的推论有： </p> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml"><i>e</i><sup><i>aX</i></sup><i>e</i><sup><i>bX</i></sup> = <i>e</i><sup>(<i>a</i> + <i>b</i>)<i>X</i></sup></span></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml"><i>e</i><sup><i>X</i></sup><i>e</i><sup>−<i>X</i></sup> = <i>I</i></span></span></li></ul> <div class="mw-heading mw-heading3"><h3 id="线性微分方程"><span id=".E7.BA.BF.E6.80.A7.E5.BE.AE.E5.88.86.E6.96.B9.E7.A8.8B"></span>线性微分方程</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=3" title="编辑章节：线性微分方程"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>矩阵指数的一个重要性，是它可以用来解<a href="/wiki/%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="微分方程">微分方程</a>。从(1)可知，以下微分方程 </p> <dl><dd><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 {\frac {d}{dt}}y(t)=Ay(t),\quad y(0)=y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}y(t)=Ay(t),\quad y(0)=y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ad824d46562423b5760ae54a1dd278f2ba2cbb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.118ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}y(t)=Ay(t),\quad y(0)=y_{0}}"></span></dd></dl> <p>其中<i>A</i>是矩阵，具有解 </p> <dl><dd><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 y(t)=e^{At}y_{0}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mi>t</mi> </mrow> </msup> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)=e^{At}y_{0}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f141eeeb4fcdceb5c5b824869161f0e1591edfc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.819ex; height:3.176ex;" alt="{\displaystyle y(t)=e^{At}y_{0}\ }"></span></dd></dl> <p>矩阵指数也可以用来解非齐次方程： </p> <dl><dd><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 {\frac {d}{dt}}y(t)=Ay(t)+z(t),\quad y(0)=y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>z</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}y(t)=Ay(t)+z(t),\quad y(0)=y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f583f6e31f827d640f7883230ee7559cdc7c9a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:34.695ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}y(t)=Ay(t)+z(t),\quad y(0)=y_{0}}"></span></dd></dl> <p>参见<a href="#应用">以下的例子</a>。 </p><p>当<i>A</i>不是常数时，以下形式的微分方程没有闭式解： </p> <dl><dd><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 {\frac {d}{dt}}y(t)=A(t)\,y(t),\quad y(0)=y_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>y</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}y(t)=A(t)\,y(t),\quad y(0)=y_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21def6f44150ac3685410e98d09bdcf28b872822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.154ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}y(t)=A(t)\,y(t),\quad y(0)=y_{0}}"></span></dd></dl> <p>但<a href="/w/index.php?title=%E9%A9%AC%E6%A0%BC%E5%8A%AA%E6%96%AF%E7%BA%A7%E6%95%B0&action=edit&redlink=1" class="new" title="马格努斯级数（页面不存在）">马格努斯级数</a>可以给出无穷级数形式的解。 </p> <div class="mw-heading mw-heading3"><h3 id="矩阵指数的行列式"><span id=".E7.9F.A9.E9.98.B5.E6.8C.87.E6.95.B0.E7.9A.84.E8.A1.8C.E5.88.97.E5.BC.8F"></span>矩阵指数的行列式</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=4" title="编辑章节：矩阵指数的行列式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>根据<a href="/wiki/%E9%9B%85%E5%8F%AF%E6%AF%94%E5%85%AC%E5%BC%8F" title="雅可比公式">雅可比公式</a>，对任意复矩阵，下列<a href="/w/index.php?title=%E8%BF%B9%E7%AD%89%E5%BC%8F&action=edit&redlink=1" class="new" title="迹等式（页面不存在）">迹等式</a>成立：<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; background-color: #ECFCF4; text-align: center; display: table"> <p><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 \det(e^{A})=e^{\operatorname {tr} (A)}~}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(e^{A})=e^{\operatorname {tr} (A)}~}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70504a50d56aa84f6a2a7ce2f5df19b62b0e7db4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.378ex; height:3.343ex;" alt="{\displaystyle \det(e^{A})=e^{\operatorname {tr} (A)}~}"></span> </p> </div> <p>除了提供一种额外的计算工具，这个等式还表明矩阵指数总是<a href="/wiki/%E5%8F%AF%E9%80%86%E7%9F%A9%E9%98%B5" class="mw-redirect" title="可逆矩阵">可逆矩阵</a>。这点可以如下证明：因为上述等式的右边恒不等于0，所以左边<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml">det(<i>e<sup>A</sup></i>) ≠ 0</span></span>，从而<link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r58896141"><span class="serif"><span class="texhtml"><i>e<sup>A</sup></i></span></span>必可逆。 </p> <div class="mw-heading mw-heading3"><h3 id="指数相加"><span id=".E6.8C.87.E6.95.B0.E7.9B.B8.E5.8A.A0"></span>指数相加</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=5" title="编辑章节：指数相加"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>我们知道，对于任何实数（标量）<i>x</i>和<i>y</i>，指数函数都满足公式<i>e</i><sup><i>x</i> + <i>y</i></sup> = <i>e</i><sup><i>x</i></sup><i>e</i><sup><i>y</i></sup>。类似的等式对于可交换矩阵也成立：如果矩阵<i>X</i>和<i>Y</i>是可交换的（即<i>XY</i> = <i>YX</i>），则： </p> <dl><dd><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 e^{X+Y}=e^{X}e^{Y}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X+Y}=e^{X}e^{Y}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4534e24f921891c79e87278add9af4c4b2d8e96e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.213ex; height:2.676ex;" alt="{\displaystyle e^{X+Y}=e^{X}e^{Y}\ }"></span></dd></dl> <p>但是，如果它们不是可交换的，则以上的等式不一定成立。 </p><p>这个命题反过来不成立：<i>e</i><sup><i>X</i>+<i>Y</i></sup>=<i>e</i><sup><i>X</i></sup><i>e</i><sup><i>Y</i></sup>并不一定就意味着<i>X</i>和<i>Y</i>是可交换的。但是，如果<i>X</i>和<i>Y</i>只含有<a href="/wiki/%E4%BB%A3%E6%95%B0%E6%95%B0" class="mw-redirect" title="代数数">代数数</a>，而且它们的大小至少为2×2，则反过来也成立<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>。 </p><p><i>X</i>和<i>Y</i>不可交换的情况可以用以下方法计算： </p> <div class="mw-heading mw-heading3"><h3 id="李乘积公式"><span id=".E6.9D.8E.E4.B9.98.E7.A7.AF.E5.85.AC.E5.BC.8F"></span>李乘积公式</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=6" title="编辑章节：李乘积公式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>即使<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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>和<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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>不可交换，<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 e^{X+Y}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X+Y}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/acc4f22a90b0b91696beba01b7e5833f32d96ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.248ex; height:2.676ex;" alt="{\displaystyle e^{X+Y}}"></span>可以用<a href="/w/index.php?title=%E6%9D%8E%E4%B9%98%E7%A7%AF%E5%85%AC%E5%BC%8F&action=edit&redlink=1" class="new" title="李乘积公式（页面不存在）">李乘积公式</a>来计算<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><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 e^{X+Y}=\lim _{n\rightarrow \infty }(e^{X/n}e^{Y/n})^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </msup> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X+Y}=\lim _{n\rightarrow \infty }(e^{X/n}e^{Y/n})^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e47e5a0325e36be085b5951b76697568f9eecca0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:24.549ex; height:4.343ex;" alt="{\displaystyle e^{X+Y}=\lim _{n\rightarrow \infty }(e^{X/n}e^{Y/n})^{n}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="贝克尔-坎贝尔-豪斯多夫公式"><span id=".E8.B4.9D.E5.85.8B.E5.B0.94-.E5.9D.8E.E8.B4.9D.E5.B0.94-.E8.B1.AA.E6.96.AF.E5.A4.9A.E5.A4.AB.E5.85.AC.E5.BC.8F"></span>贝克尔-坎贝尔-豪斯多夫公式</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=7" title="编辑章节：贝克尔-坎贝尔-豪斯多夫公式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>从另一个方向讲，如果<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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>和<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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>是元素足够小（但不一定可交换）的矩阵，我们有： </p> <dl><dd><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 e^{X}e^{Y}=e^{Z}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Y</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>Z</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X}e^{Y}=e^{Z}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a94ba143584ba4eaf7ce2fb6cfa354e509d4c12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.888ex; height:2.676ex;" alt="{\displaystyle e^{X}e^{Y}=e^{Z}}"></span></dd></dl> <p>其中<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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}"></span>可以通过<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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>和<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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>的交换子的级数（<a href="/w/index.php?title=%E8%B4%9D%E5%85%8B%E5%B0%94-%E5%9D%8E%E8%B4%9D%E5%B0%94-%E8%B1%AA%E6%96%AF%E5%A4%9A%E5%A4%AB%E5%85%AC%E5%BC%8F&action=edit&redlink=1" class="new" title="贝克尔-坎贝尔-豪斯多夫公式（页面不存在）">贝克尔-坎贝尔-豪斯多夫公式</a>）来计算：<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> <dl><dd><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 Z=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mi>X</mi> <mo>+</mo> <mi>Y</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>12</mn> </mfrac> </mrow> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mo stretchy="false">[</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]+\cdots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d84e2fc66944525ee2d8fb0c268595ac226160c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.246ex; height:5.176ex;" alt="{\displaystyle Z=X+Y+{\frac {1}{2}}[X,Y]+{\frac {1}{12}}[X,[X,Y]]+\cdots }"></span></dd></dl> <p>其中余项均为与<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 X}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68baa052181f707c662844a465bfeeb135e82bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}"></span>和<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 Y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/961d67d6b454b4df2301ac571808a3538b3a6d3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}"></span>相关的迭代交换子。 </p> <div class="mw-heading mw-heading3"><h3 id="指数映射"><span id=".E6.8C.87.E6.95.B0.E6.98.A0.E5.B0.84"></span>指数映射</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=8" title="编辑章节：指数映射"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>注意矩阵的指数总是<a href="/wiki/%E9%9D%9E%E5%A5%87%E5%BC%82%E6%96%B9%E9%98%B5" title="非奇异方阵">非奇异方阵</a>。<i>e</i><sup><i>X</i></sup>的<a href="/wiki/%E9%80%86%E7%9F%A9%E9%98%B5" title="逆矩阵">逆矩阵</a>由<i>e</i><sup>−<i>X</i></sup>给出。这与复数的指数总是非零的事实类似。这样，矩阵指数就给出了一个映射： </p> <dl><dd><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 \exp \colon M_{n}(\mathbb {C} )\to {\mbox{GL}}(n,\mathbb {C} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>:</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>GL</mtext> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \colon M_{n}(\mathbb {C} )\to {\mbox{GL}}(n,\mathbb {C} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0ec994e7437c7b5a857927ccc2e7ffc9457d1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.354ex; height:2.843ex;" alt="{\displaystyle \exp \colon M_{n}(\mathbb {C} )\to {\mbox{GL}}(n,\mathbb {C} )}"></span></dd></dl> <p>这是从所有<i>n</i>×<i>n</i>矩阵的空间到<a href="/wiki/%E4%B8%80%E8%88%AC%E7%BA%BF%E6%80%A7%E7%BE%A4" title="一般线性群">一般线性群</a>（所有非奇异方阵所组成的群）的映射。实际上，这个映射是<a href="/wiki/%E6%BB%A1%E5%B0%84" title="满射">满射</a>，就是说每一个非奇异方阵都可以写成某个矩阵的指数。<a href="/wiki/%E7%9F%A9%E9%98%B5%E5%AF%B9%E6%95%B0" title="矩阵对数">矩阵对数</a>就是这个映射的逆映射。 </p><p>对于任何两个矩阵<i>X</i>和<i>Y</i>，我们有： </p> <dl><dd><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 \|e^{X+Y}-e^{X}\|\leq \|Y\|e^{\|X\|}e^{\|Y\|}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo>+</mo> <mi>Y</mi> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo fence="false" stretchy="false">‖</mo> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo fence="false" stretchy="false">‖</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>X</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">‖</mo> <mi>Y</mi> <mo fence="false" stretchy="false">‖</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|e^{X+Y}-e^{X}\|\leq \|Y\|e^{\|X\|}e^{\|Y\|}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db4e0655c61b0c94c1e9061c17a82687cae2b533" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.899ex; height:3.343ex;" alt="{\displaystyle \|e^{X+Y}-e^{X}\|\leq \|Y\|e^{\|X\|}e^{\|Y\|}}"></span></dd></dl> <p>其中|| · ||表示任意的<a href="/wiki/%E7%9F%A9%E9%98%B5%E8%8C%83%E6%95%B0" class="mw-redirect" title="矩阵范数">矩阵范数</a>。从中可以推出，指数映射在<i>M</i><sub><i>n</i></sub>(<b>C</b>)的<a href="/wiki/%E7%B4%A7%E9%9B%86" class="mw-redirect" title="紧集">紧子集</a>内是<a href="/wiki/%E8%BF%9E%E7%BB%AD" class="mw-redirect" title="连续">连续</a>和<a href="/wiki/%E5%88%A9%E6%99%AE%E5%B8%8C%E8%8C%A8%E8%BF%9E%E7%BB%AD" class="mw-redirect" title="利普希茨连续">利普希茨连续</a>的。 </p><p>以下的映射 </p> <dl><dd><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 t\mapsto e^{tX},\qquad t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">↦</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo>,</mo> <mspace width="2em" /> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\mapsto e^{tX},\qquad t\in \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c8cfd3a6dbdcf5b4fefb10c39c679dd72db266" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.801ex; height:3.009ex;" alt="{\displaystyle t\mapsto e^{tX},\qquad t\in \mathbb {R} }"></span></dd></dl> <p>定义了一般线性群中的一条<a href="/wiki/%E5%85%89%E6%BB%91" class="mw-redirect" title="光滑">光滑</a>曲线，当<i>t</i> = 0时穿过单位元。实际上，这给出了一般线性群的一个<a href="/wiki/%E5%8D%95%E5%8F%82%E6%95%B0%E5%AD%90%E7%BE%A4" class="mw-redirect" title="单参数子群">单参数子群</a>，这是由于： </p> <dl><dd><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 e^{tX}e^{sX}=e^{(t+s)X}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>X</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>X</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{tX}e^{sX}=e^{(t+s)X}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79f2847683a16ff9e195497db106ad313500b3c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:17.114ex; height:2.843ex;" alt="{\displaystyle e^{tX}e^{sX}=e^{(t+s)X}\ }"></span></dd></dl> <p>这条曲线在点<i>t</i>的导数（或<a href="/wiki/%E6%9B%B2%E7%BA%BF%E7%9A%84%E5%BE%AE%E5%88%86%E5%87%A0%E4%BD%95" title="曲线的微分几何">切向量</a>）由以下等式给出： </p> <dl><dd><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 {\frac {d}{dt}}e^{tX}=Xe^{tX}\qquad (1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mo>=</mo> <mi>X</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>X</mi> </mrow> </msup> <mspace width="2em" /> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}e^{tX}=Xe^{tX}\qquad (1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0e79c69d8e432cd4fc7e2aa2978c9f936a8a232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:22.206ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}e^{tX}=Xe^{tX}\qquad (1)}"></span></dd></dl> <p><i>t</i> = 0时的导数就是矩阵<i>X</i>，所以我们可以说，<i>X</i>是这个单参数子群的推广。 </p><p>更加一般地： </p> <dl><dd><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 {\frac {d}{dt}}e^{X(t)}=\int _{0}^{1}e^{(1-\alpha )X(t)}{\frac {dX(t)}{dt}}e^{\alpha X(t)}\,d\alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> <mo stretchy="false">)</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>X</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}e^{X(t)}=\int _{0}^{1}e^{(1-\alpha )X(t)}{\frac {dX(t)}{dt}}e^{\alpha X(t)}\,d\alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c617ad8e8779127d94bc9edf8313da4880c32b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:38.917ex; height:6.176ex;" alt="{\displaystyle {\frac {d}{dt}}e^{X(t)}=\int _{0}^{1}e^{(1-\alpha )X(t)}{\frac {dX(t)}{dt}}e^{\alpha X(t)}\,d\alpha }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="矩阵指数的计算"><span id=".E7.9F.A9.E9.98.B5.E6.8C.87.E6.95.B0.E7.9A.84.E8.AE.A1.E7.AE.97"></span>矩阵指数的计算</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=9" title="编辑章节：矩阵指数的计算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>找到可靠而准确的方法来计算矩阵指数是很困难的，这仍然是目前数学和数值分析领域的一个重要研究课题。<a href="/wiki/Matlab" class="mw-redirect" title="Matlab">Matlab</a>、<a href="/wiki/GNU_Octave" title="GNU Octave">GNU Octave</a>和<a href="/wiki/SciPy" title="SciPy">SciPy</a>都使用<a href="/wiki/%E5%B8%95%E5%BE%B7%E8%BF%91%E4%BC%BC" title="帕德近似">帕德近似</a>。<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><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><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> 在本节中，我们讨论了原则上适用于任何矩阵的方法，并且可以对小矩阵进行显式处理。<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> 随后的章节描述了适合对大矩阵进行数值评估的方法。 </p> <div class="mw-heading mw-heading3"><h3 id="可对角化矩阵"><span id=".E5.8F.AF.E5.AF.B9.E8.A7.92.E5.8C.96.E7.9F.A9.E9.98.B5"></span>可对角化矩阵</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=10" title="编辑章节：可对角化矩阵"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如果矩阵是<a href="/wiki/%E5%AF%B9%E8%A7%92%E7%9F%A9%E9%98%B5" class="mw-redirect" title="对角矩阵">对角</a>的： </p> <dl><dd><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={\begin{bmatrix}a_{1}&0&\ldots &0\\0&a_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &a_{n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}a_{1}&0&\ldots &0\\0&a_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &a_{n}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa9298c4b34b1894cb76f0e2cfe6452ab611f725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:25.657ex; height:14.176ex;" alt="{\displaystyle A={\begin{bmatrix}a_{1}&0&\ldots &0\\0&a_{2}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &a_{n}\end{bmatrix}}}"></span></dd></dl> <p>则把主对角线上的所有元素取指数，就是原矩阵的指数： </p> <dl><dd><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 e^{A}={\begin{bmatrix}e^{a_{1}}&0&\ldots &0\\0&e^{a_{2}}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &e^{a_{n}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msup> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>…</mo> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{A}={\begin{bmatrix}e^{a_{1}}&0&\ldots &0\\0&e^{a_{2}}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &e^{a_{n}}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6325ed81955d3a99b0abfed0a386ca11b18fa93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:28.629ex; height:14.176ex;" alt="{\displaystyle e^{A}={\begin{bmatrix}e^{a_{1}}&0&\ldots &0\\0&e^{a_{2}}&\ldots &0\\\vdots &\vdots &\ddots &\vdots \\0&0&\ldots &e^{a_{n}}\end{bmatrix}}}"></span></dd></dl> <p>这也允许了我们计算<a href="/wiki/%E5%8F%AF%E5%AF%B9%E8%A7%92%E5%8C%96%E7%9F%A9%E9%98%B5" title="可对角化矩阵">可对角化矩阵</a>的指数。如果<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=UDU^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>U</mi> <mi>D</mi> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=UDU^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49e76df72922aa8c7b75b70f9bb83d1115bb3db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.722ex; height:2.676ex;" alt="{\displaystyle A=UDU^{-1}}"></span>，且<i>D</i>是对角矩阵，则<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 e^{A}=Ue^{D}U^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <mo>=</mo> <mi>U</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> </mrow> </msup> <msup> <mi>U</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{A}=Ue^{D}U^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68d9cb9147c72ae530bfdbcb4b79ac6ec4594e93" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.28ex; height:2.676ex;" alt="{\displaystyle e^{A}=Ue^{D}U^{-1}}"></span>。用<a href="/w/index.php?title=%E8%A5%BF%E5%B0%94%E7%BB%B4%E6%96%AF%E7%89%B9%E5%85%AC%E5%BC%8F&action=edit&redlink=1" class="new" title="西尔维斯特公式（页面不存在）">西尔维斯特公式</a>，也可以得到相同的结果。 </p> <div class="mw-heading mw-heading3"><h3 id="幂零矩阵"><span id=".E5.B9.82.E9.9B.B6.E7.9F.A9.E9.98.B5"></span>幂零矩阵</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=11" title="编辑章节：幂零矩阵"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>如果对于某个整数<i>q</i>，有<i>N</i><sup><i>q</i></sup> = 0，则矩阵<i>N</i>称为<a href="/wiki/%E5%B9%82%E9%9B%B6%E7%9F%A9%E9%98%B5" title="幂零矩阵">幂零矩阵</a>。在这种情况下，矩阵指数<i>e</i><sup><i>N</i></sup>可以直接从级数展开式来计算，这是因为级数在有限个项后就终止了： </p> <dl><dd><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 e^{N}=I+N+{\frac {1}{2}}N^{2}+{\frac {1}{6}}N^{3}+\cdots +{\frac {1}{(q-1)!}}N^{q-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mo>=</mo> <mi>I</mi> <mo>+</mo> <mi>N</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>6</mn> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mo stretchy="false">(</mo> <mi>q</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>!</mo> </mrow> </mfrac> </mrow> <msup> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{N}=I+N+{\frac {1}{2}}N^{2}+{\frac {1}{6}}N^{3}+\cdots +{\frac {1}{(q-1)!}}N^{q-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2435488970a691ed92b10d74eaad22e7aa3b47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.962ex; height:6.009ex;" alt="{\displaystyle e^{N}=I+N+{\frac {1}{2}}N^{2}+{\frac {1}{6}}N^{3}+\cdots +{\frac {1}{(q-1)!}}N^{q-1}}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="推广"><span id=".E6.8E.A8.E5.B9.BF"></span>推广</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=12" title="编辑章节：推广"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>当矩阵<i>X</i>的<a href="/wiki/%E6%9C%80%E5%B0%8F%E5%A4%9A%E9%A0%85%E5%BC%8F" class="mw-redirect" title="最小多項式">最小多项式</a>可以分解为一次多项式的积时，它就可以表示为以下的和： </p> <dl><dd><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 X=A+N\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mi>A</mi> <mo>+</mo> <mi>N</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=A+N\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14d320d9862480562243159f6c5888544f6573ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.306ex; height:2.343ex;" alt="{\displaystyle X=A+N\ }"></span></dd></dl> <p>其中： </p> <ul><li><i>A</i>是可对角化矩阵；</li> <li><i>N</i>是幂零矩阵；</li> <li><i>A</i>与<i>N</i>是可交换的（也就是说， <i>AN</i> = <i>NA</i>）。</li></ul> <p>这称为<a href="/w/index.php?title=Dunford%E5%88%86%E8%A7%A3&action=edit&redlink=1" class="new" title="Dunford分解（页面不存在）">Dunford分解</a>。 </p><p>这就是说，我们可以通过化为前两种情况，来计算<i>X</i>的指数： </p> <dl><dd><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 e^{X}=e^{A+N}=e^{A}e^{N}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> <mo>+</mo> <mi>N</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X}=e^{A+N}=e^{A}e^{N}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea913c5be69226fab98aef8d9f5cc6809f6b8b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:20.103ex; height:2.676ex;" alt="{\displaystyle e^{X}=e^{A+N}=e^{A}e^{N}\ }"></span></dd></dl> <p>注意为了让最后一步成立， <i>A</i>和<i>N</i>必须是可交换的。 </p><p>另外一个密切相关的方法，是利用<i>X</i>的<a href="/wiki/%E8%8B%A5%E5%B0%94%E5%BD%93%E6%A0%87%E5%87%86%E5%9E%8B" title="若尔当标准型">若尔当标准型</a>。假设<i>X</i> = <i>PJP</i><sup> −1</sup>，其中<i>J</i>是<i>X</i>的<a href="/wiki/%E8%8B%A5%E5%B0%94%E5%BD%93%E6%A0%87%E5%87%86%E5%9E%8B" title="若尔当标准型">若尔当标准型</a>。那么： </p> <dl><dd><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 e^{X}=Pe^{J}P^{-1}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>X</mi> </mrow> </msup> <mo>=</mo> <mi>P</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msup> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{X}=Pe^{J}P^{-1}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c10b6fb2bd542069682c44b17b8a0ec2aee22eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.651ex; height:2.676ex;" alt="{\displaystyle e^{X}=Pe^{J}P^{-1}\ }"></span></dd></dl> <p>另外，由于 </p> <dl><dd><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 J=J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊕</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8cf188ef7644c9ea81b79ed2887e20b1034c30f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.439ex; height:3.009ex;" alt="{\displaystyle J=J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n})}"></span></dd></dl> <dl><dd><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 {\begin{aligned}e^{J}&{}=\exp {\big (}J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n}){\big )}\\&{}=\exp {\big (}J_{a_{1}}(\lambda _{1}){\big )}\oplus \exp {\big (}J_{a_{2}}(\lambda _{2}){\big )}\oplus \cdots \oplus \exp {\big (}J_{a_{k}}(\lambda _{k}){\big )}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>J</mi> </mrow> </msup> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊕</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>⊕</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> <mo>⊕</mo> <mo>⋯</mo> <mo>⊕</mo> <mi>exp</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}e^{J}&{}=\exp {\big (}J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n}){\big )}\\&{}=\exp {\big (}J_{a_{1}}(\lambda _{1}){\big )}\oplus \exp {\big (}J_{a_{2}}(\lambda _{2}){\big )}\oplus \cdots \oplus \exp {\big (}J_{a_{k}}(\lambda _{k}){\big )}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/653f77430fc3fdc04019d162bf9d07409da772aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:58.048ex; height:6.843ex;" alt="{\displaystyle {\begin{aligned}e^{J}&{}=\exp {\big (}J_{a_{1}}(\lambda _{1})\oplus J_{a_{2}}(\lambda _{2})\oplus \cdots \oplus J_{a_{n}}(\lambda _{n}){\big )}\\&{}=\exp {\big (}J_{a_{1}}(\lambda _{1}){\big )}\oplus \exp {\big (}J_{a_{2}}(\lambda _{2}){\big )}\oplus \cdots \oplus \exp {\big (}J_{a_{k}}(\lambda _{k}){\big )}\end{aligned}}}"></span></dd></dl> <p>因此，我们只需要知道怎样计算<a href="/wiki/%E8%8B%A5%E5%B0%94%E5%BD%93%E7%9F%A9%E9%98%B5" title="若尔当矩阵">若尔当块</a>的矩阵指数。但是，每一个若尔当块都具有形式 </p> <dl><dd><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 J_{a}(\lambda )=\lambda I+N\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>a</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>I</mi> <mo>+</mo> <mi>N</mi> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{a}(\lambda )=\lambda I+N\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d8445cc7da265d570fb213132b7293f04b8c7f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.667ex; height:2.843ex;" alt="{\displaystyle J_{a}(\lambda )=\lambda I+N\ }"></span></dd></dl> <p>其中<i>N</i>是幂零矩阵。则这个区块的矩阵指数由下式给出： </p> <dl><dd><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 e^{\lambda I+N}=e^{\lambda }e^{N}\ }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>I</mi> <mo>+</mo> <mi>N</mi> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> </mrow> </msup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msup> <mtext> </mtext> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{\lambda I+N}=e^{\lambda }e^{N}\ }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50e92b0c4cb5e42e9503951016bc770fe933a16d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.568ex; height:2.676ex;" alt="{\displaystyle e^{\lambda I+N}=e^{\lambda }e^{N}\ }"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="计算"><span id=".E8.AE.A1.E7.AE.97"></span>计算</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=13" title="编辑章节：计算"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>假设我们想要计算以下矩阵的指数。 </p> <dl><dd><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={\begin{bmatrix}21&17&6\\-5&-1&-6\\4&4&16\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>21</mn> </mtd> <mtd> <mn>17</mn> </mtd> <mtd> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>5</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>6</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>16</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B={\begin{bmatrix}21&17&6\\-5&-1&-6\\4&4&16\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77800dd82aed570bc85a8b69ab82a2ab11b5cb02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:22.272ex; height:9.176ex;" alt="{\displaystyle B={\begin{bmatrix}21&17&6\\-5&-1&-6\\4&4&16\end{bmatrix}}}"></span></dd></dl> <p>它的若尔当型为： </p> <dl><dd><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 J=P^{-1}BP={\begin{bmatrix}4&0&0\\0&16&1\\0&0&16\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>B</mi> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>16</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>16</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=P^{-1}BP={\begin{bmatrix}4&0&0\\0&16&1\\0&0&16\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8aa6ab48a4792fd9a195e46d01978e141bd78fde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:29.642ex; height:9.176ex;" alt="{\displaystyle J=P^{-1}BP={\begin{bmatrix}4&0&0\\0&16&1\\0&0&16\end{bmatrix}}}"></span></dd></dl> <p>其中矩阵<i>P</i>由下式给出： </p> <dl><dd><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 P={\begin{bmatrix}-{\frac {1}{4}}&2&{\frac {5}{4}}\\{\frac {1}{4}}&-2&-{\frac {1}{4}}\\0&4&0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P={\begin{bmatrix}-{\frac {1}{4}}&2&{\frac {5}{4}}\\{\frac {1}{4}}&-2&-{\frac {1}{4}}\\0&4&0\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbac33e43f19b50c7c2890e96a75369f1ccbbc5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:23.244ex; height:10.843ex;" alt="{\displaystyle P={\begin{bmatrix}-{\frac {1}{4}}&2&{\frac {5}{4}}\\{\frac {1}{4}}&-2&-{\frac {1}{4}}\\0&4&0\end{bmatrix}}}"></span></dd></dl> <p>我们首先来计算exp(<i>J</i>)。我们有： </p> <dl><dd><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 J=J_{1}(4)\oplus J_{2}(16)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>⊕</mo> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=J_{1}(4)\oplus J_{2}(16)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88f45865d147222d8b16ef38760e91c30aded3dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.205ex; height:2.843ex;" alt="{\displaystyle J=J_{1}(4)\oplus J_{2}(16)}"></span></dd></dl> <p>1×1矩阵的指数仅仅是该矩阵的元素的指数，因此exp(<i>J</i><sub>1</sub>(4)) = [<i>e</i><sup>4</sup>]。<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 J_{2}(16)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{2}(16)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aaa09c8e85e5a34e836ced4c142e546cf26e9f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.479ex; height:2.843ex;" alt="{\displaystyle J_{2}(16)}"></span>的指数可以用以上提到的公式exp(λ<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 I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"></span>+<i>N</i>) = <i>e</i><sup>λ</sup> exp(<i>N</i>)来算出： </p> <dl><dd><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 \exp \left({\begin{bmatrix}16&1\\0&16\end{bmatrix}}\right)=e^{16}\exp \left({\begin{bmatrix}0&1\\0&0\end{bmatrix}}\right)=e^{16}\left({\begin{bmatrix}1&0\\0&1\end{bmatrix}}+{\begin{bmatrix}0&1\\0&0\end{bmatrix}}+{1 \over 2!}{\begin{bmatrix}0&0\\0&0\end{bmatrix}}+\cdots \right)={\begin{bmatrix}e^{16}&e^{16}\\0&e^{16}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>16</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>16</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mo>!</mo> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mo>⋯</mo> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp \left({\begin{bmatrix}16&1\\0&16\end{bmatrix}}\right)=e^{16}\exp \left({\begin{bmatrix}0&1\\0&0\end{bmatrix}}\right)=e^{16}\left({\begin{bmatrix}1&0\\0&1\end{bmatrix}}+{\begin{bmatrix}0&1\\0&0\end{bmatrix}}+{1 \over 2!}{\begin{bmatrix}0&0\\0&0\end{bmatrix}}+\cdots \right)={\begin{bmatrix}e^{16}&e^{16}\\0&e^{16}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e45d707540e443bb8e8d627dedb214acc879469c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:100.29ex; height:6.176ex;" alt="{\displaystyle \exp \left({\begin{bmatrix}16&1\\0&16\end{bmatrix}}\right)=e^{16}\exp \left({\begin{bmatrix}0&1\\0&0\end{bmatrix}}\right)=e^{16}\left({\begin{bmatrix}1&0\\0&1\end{bmatrix}}+{\begin{bmatrix}0&1\\0&0\end{bmatrix}}+{1 \over 2!}{\begin{bmatrix}0&0\\0&0\end{bmatrix}}+\cdots \right)={\begin{bmatrix}e^{16}&e^{16}\\0&e^{16}\end{bmatrix}}}"></span></dd></dl> <p>因此，原矩阵<i>B</i>的指数为： </p> <dl><dd><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 \exp(B)=P\exp(J)P^{-1}=P{\begin{bmatrix}e^{4}&0&0\\0&e^{16}&e^{16}\\0&0&e^{16}\end{bmatrix}}P^{-1}={1 \over 4}{\begin{bmatrix}13e^{16}-e^{4}&13e^{16}-5e^{4}&2e^{16}-2e^{4}\\-9e^{16}+e^{4}&-9e^{16}+5e^{4}&-2e^{16}+2e^{4}\\16e^{16}&16e^{16}&4e^{16}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>P</mi> <mi>exp</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>J</mi> <mo stretchy="false">)</mo> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <msup> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>4</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>13</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mn>13</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>−</mo> <mn>5</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>9</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>9</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>+</mo> <mn>5</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> <mo>+</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>16</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> <mtd> <mn>16</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> <mtd> <mn>4</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>16</mn> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \exp(B)=P\exp(J)P^{-1}=P{\begin{bmatrix}e^{4}&0&0\\0&e^{16}&e^{16}\\0&0&e^{16}\end{bmatrix}}P^{-1}={1 \over 4}{\begin{bmatrix}13e^{16}-e^{4}&13e^{16}-5e^{4}&2e^{16}-2e^{4}\\-9e^{16}+e^{4}&-9e^{16}+5e^{4}&-2e^{16}+2e^{4}\\16e^{16}&16e^{16}&4e^{16}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59ed176c8490bda2fa2ef74e2336ec5cb43c652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:97.541ex; height:9.509ex;" alt="{\displaystyle \exp(B)=P\exp(J)P^{-1}=P{\begin{bmatrix}e^{4}&0&0\\0&e^{16}&e^{16}\\0&0&e^{16}\end{bmatrix}}P^{-1}={1 \over 4}{\begin{bmatrix}13e^{16}-e^{4}&13e^{16}-5e^{4}&2e^{16}-2e^{4}\\-9e^{16}+e^{4}&-9e^{16}+5e^{4}&-2e^{16}+2e^{4}\\16e^{16}&16e^{16}&4e^{16}\end{bmatrix}}}"></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="应用"><span id=".E5.BA.94.E7.94.A8"></span>应用</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=14" title="编辑章节：应用"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="线性微分方程_2"><span id=".E7.BA.BF.E6.80.A7.E5.BE.AE.E5.88.86.E6.96.B9.E7.A8.8B_2"></span>线性微分方程</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=15" title="编辑章节：线性微分方程"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>矩阵指数在解<a href="/wiki/%E7%BA%BF%E6%80%A7%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B" title="线性微分方程">线性微分方程</a>时十分有用。前面曾提到，以下形式的微分方程 </p> <dl><dd><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 \mathbf {y} '=C\mathbf {y} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>′</mo> </msup> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} '=C\mathbf {y} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba6a7458830e945347ca011f19117007a75b53be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.371ex; height:2.843ex;" alt="{\displaystyle \mathbf {y} '=C\mathbf {y} }"></span></dd></dl> <p>具有解<i>e</i><sup>C<i>t</i></sup><i>y(0)</i>。如果我们考虑以下向量 </p> <dl><dd><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 \mathbf {y} (t)={\begin{bmatrix}y_{1}(t)\\\vdots \\y_{n}(t)\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} (t)={\begin{bmatrix}y_{1}(t)\\\vdots \\y_{n}(t)\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3649644dcd622af5083b23f56aa3c8c3984584b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:16.017ex; height:10.843ex;" alt="{\displaystyle \mathbf {y} (t)={\begin{bmatrix}y_{1}(t)\\\vdots \\y_{n}(t)\end{bmatrix}}}"></span></dd></dl> <p>我们就可以把线性微分方程表示为： </p> <dl><dd><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 \mathbf {y} '(t)=A\mathbf {y} (t)+\mathbf {b} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} '(t)=A\mathbf {y} (t)+\mathbf {b} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13dbe438d89d57d76778ce0a0eca47c81814332f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.621ex; height:3.009ex;" alt="{\displaystyle \mathbf {y} '(t)=A\mathbf {y} (t)+\mathbf {b} (t)}"></span></dd></dl> <p>如果我们作一个猜想，把两边乘以一个<a href="/wiki/%E7%A7%AF%E5%88%86%E5%9B%A0%E5%AD%90" title="积分因子">积分因子</a> <i>e</i><sup>−<i>At</i></sup>，便得到： </p> <dl><dd><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 e^{-At}\mathbf {y} '-e^{-At}A\mathbf {y} =e^{-At}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>A</mi> <mi>t</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>′</mo> </msup> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>A</mi> <mi>t</mi> </mrow> </msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>A</mi> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{-At}\mathbf {y} '-e^{-At}A\mathbf {y} =e^{-At}\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d9aa5b78e6659a2d32d4decef1005369ba7a946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:25.936ex; height:3.009ex;" alt="{\displaystyle e^{-At}\mathbf {y} '-e^{-At}A\mathbf {y} =e^{-At}\mathbf {b} }"></span></dd> <dd><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 {\frac {d}{dt}}(e^{-At}\mathbf {y} )=e^{-At}\mathbf {b} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>A</mi> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>A</mi> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}(e^{-At}\mathbf {y} )=e^{-At}\mathbf {b} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/402beab1773bf0ebddac7c9e501d0a421683c8e1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.537ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}(e^{-At}\mathbf {y} )=e^{-At}\mathbf {b} }"></span></dd></dl> <p>如果我们可以计算<i>e</i><sup><i>At</i></sup>，那么就得到了微分方程的解。 </p> <div class="mw-heading mw-heading4"><h4 id="例子（齐次）"><span id=".E4.BE.8B.E5.AD.90.EF.BC.88.E9.BD.90.E6.AC.A1.EF.BC.89"></span>例子（齐次）</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=16" title="编辑章节：例子（齐次）"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>假设我们有以下的微分方程组： </p> <dl><dd><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 {\begin{matrix}x'&=&2x&-y&+z\\y'&=&&3y&-1z\\z'&=&2x&+y&+3z\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mo>+</mo> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd /> <mtd> <mn>3</mn> <mi>y</mi> </mtd> <mtd> <mo>−</mo> <mn>1</mn> <mi>z</mi> </mtd> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mo>+</mo> <mi>y</mi> </mtd> <mtd> <mo>+</mo> <mn>3</mn> <mi>z</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x'&=&2x&-y&+z\\y'&=&&3y&-1z\\z'&=&2x&+y&+3z\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6d4792379d8eef0156649a4d6d008c304794807" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:23.379ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}x'&=&2x&-y&+z\\y'&=&&3y&-1z\\z'&=&2x&+y&+3z\end{matrix}}}"></span></dd></dl> <p>相关的矩阵为： </p> <dl><dd><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 M={\begin{bmatrix}2&-1&1\\0&3&-1\\2&1&3\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\begin{bmatrix}2&-1&1\\0&3&-1\\2&1&3\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e69a49c6ba7d4a025867a570b62b3d826b4cc3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.142ex; height:9.176ex;" alt="{\displaystyle M={\begin{bmatrix}2&-1&1\\0&3&-1\\2&1&3\end{bmatrix}}}"></span></dd></dl> <p>在以上的例子中，我们计算了矩阵指数 </p> <dl><dd><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 e^{tM}={\begin{bmatrix}2e^{t}-2te^{2t}&-2te^{2t}&0\\-2e^{t}+2(t+1)e^{2t}&2(t+1)e^{2t}&0\\2te^{2t}&2te^{2t}&2e^{t}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>M</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{tM}={\begin{bmatrix}2e^{t}-2te^{2t}&-2te^{2t}&0\\-2e^{t}+2(t+1)e^{2t}&2(t+1)e^{2t}&0\\2te^{2t}&2te^{2t}&2e^{t}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9adacee2be390d8c415b59401f7651aeb47efbd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:47.116ex; height:9.843ex;" alt="{\displaystyle e^{tM}={\begin{bmatrix}2e^{t}-2te^{2t}&-2te^{2t}&0\\-2e^{t}+2(t+1)e^{2t}&2(t+1)e^{2t}&0\\2te^{2t}&2te^{2t}&2e^{t}\end{bmatrix}}}"></span></dd></dl> <p>因此微分方程组的通解为： </p> <dl><dd><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 {\begin{bmatrix}x\\y\\z\end{bmatrix}}=C_{1}{\begin{bmatrix}2e^{t}-2te^{2t}\\-2e^{t}+2(t+1)e^{2t}\\2te^{2t}\end{bmatrix}}+C_{2}{\begin{bmatrix}-2te^{2t}\\2(t+1)e^{2t}\\2te^{2t}\end{bmatrix}}+C_{3}{\begin{bmatrix}0\\0\\2e^{t}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}=C_{1}{\begin{bmatrix}2e^{t}-2te^{2t}\\-2e^{t}+2(t+1)e^{2t}\\2te^{2t}\end{bmatrix}}+C_{2}{\begin{bmatrix}-2te^{2t}\\2(t+1)e^{2t}\\2te^{2t}\end{bmatrix}}+C_{3}{\begin{bmatrix}0\\0\\2e^{t}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d182e7c15bf9145c255efacdc40ec6585c575d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:65.549ex; height:9.843ex;" alt="{\displaystyle {\begin{bmatrix}x\\y\\z\end{bmatrix}}=C_{1}{\begin{bmatrix}2e^{t}-2te^{2t}\\-2e^{t}+2(t+1)e^{2t}\\2te^{2t}\end{bmatrix}}+C_{2}{\begin{bmatrix}-2te^{2t}\\2(t+1)e^{2t}\\2te^{2t}\end{bmatrix}}+C_{3}{\begin{bmatrix}0\\0\\2e^{t}\end{bmatrix}}}"></span></dd></dl> <p>也就是说， </p> <dl><dd><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 {\begin{aligned}x&=C_{1}(2e^{t}-2te^{2t})+C_{2}(-2te^{2t})\\y&=C_{1}(-2e^{t}+2(t+1)e^{2t})+C_{2}(2(t+1)e^{2t})\\z&=(C_{1}+C_{2})(2te^{2t})+2C_{3}e^{t}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>x</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>z</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x&=C_{1}(2e^{t}-2te^{2t})+C_{2}(-2te^{2t})\\y&=C_{1}(-2e^{t}+2(t+1)e^{2t})+C_{2}(2(t+1)e^{2t})\\z&=(C_{1}+C_{2})(2te^{2t})+2C_{3}e^{t}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e55753bf089272f5c80ceddb58fd76311c7748b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.072ex; margin-bottom: -0.266ex; width:45.882ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}x&=C_{1}(2e^{t}-2te^{2t})+C_{2}(-2te^{2t})\\y&=C_{1}(-2e^{t}+2(t+1)e^{2t})+C_{2}(2(t+1)e^{2t})\\z&=(C_{1}+C_{2})(2te^{2t})+2C_{3}e^{t}\end{aligned}}}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="非齐次的情况──参数变换"><span id=".E9.9D.9E.E9.BD.90.E6.AC.A1.E7.9A.84.E6.83.85.E5.86.B5.E2.94.80.E2.94.80.E5.8F.82.E6.95.B0.E5.8F.98.E6.8D.A2"></span>非齐次的情况──参数变换</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=17" title="编辑章节：非齐次的情况──参数变换"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>对于非齐次的情况，我们可以用<a href="/wiki/%E7%A7%AF%E5%88%86%E5%9B%A0%E5%AD%90" title="积分因子">积分因子</a>的方法（类似于<a href="/w/index.php?title=%E5%8F%82%E6%95%B0%E5%8F%98%E6%8D%A2&action=edit&redlink=1" class="new" title="参数变换（页面不存在）">参数变换</a>的方法）。我们找到形为<i><b>y</b></i><sub>p</sub>(<i>t</i>) = exp(<i>tA</i>)<i><b>z</b></i>(<i>t</i>)一个特解： </p> <dl><dd><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 \mathbf {y} _{p}'=(e^{tA})'\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> <mo>′</mo> </msubsup> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}'=(e^{tA})'\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95719fe538354257c2b35c2db045a19e6f8b8d00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.546ex; height:3.343ex;" alt="{\displaystyle \mathbf {y} _{p}'=(e^{tA})'\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)}"></span></dd></dl> <dl><dd><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 =Ae^{tA}\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>A</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =Ae^{tA}\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/29ebad2eb67e3a4a1cdc9612483f475a0032d754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.679ex; height:3.176ex;" alt="{\displaystyle =Ae^{tA}\mathbf {z} (t)+e^{tA}\mathbf {z} '(t)}"></span></dd></dl> <dl><dd><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 {y} _{p}(t)+e^{tA}\mathbf {z} '(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>=</mo> <mi>A</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle =A\mathbf {y} _{p}(t)+e^{tA}\mathbf {z} '(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf16e8fb7d12738d2bcfd578c2bd079897792cda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.819ex; height:3.509ex;" alt="{\displaystyle =A\mathbf {y} _{p}(t)+e^{tA}\mathbf {z} '(t)}"></span></dd></dl> <p>为了让<i><b>y</b></i><sub>p</sub>为方程的解，必须有： </p> <dl><dd><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 e^{tA}\mathbf {z} '(t)=\mathbf {b} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e^{tA}\mathbf {z} '(t)=\mathbf {b} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce73caf9d7f5ae8b8c57cb0912cce2b59c89d61a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.896ex; height:3.176ex;" alt="{\displaystyle e^{tA}\mathbf {z} '(t)=\mathbf {b} (t)}"></span></dd></dl> <dl><dd><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 \mathbf {z} '(t)=(e^{tA})^{-1}\mathbf {b} (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo>′</mo> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {z} '(t)=(e^{tA})^{-1}\mathbf {b} (t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c95eb11e8fd08263e98afdd844635e0f03cc188" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.038ex; height:3.176ex;" alt="{\displaystyle \mathbf {z} '(t)=(e^{tA})^{-1}\mathbf {b} (t)}"></span></dd></dl> <dl><dd><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 \mathbf {z} (t)=\int _{0}^{t}e^{-uA}\mathbf {b} (u)\,du+\mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">z</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>u</mi> <mi>A</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {z} (t)=\int _{0}^{t}e^{-uA}\mathbf {b} (u)\,du+\mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd24a985f5b3e722145a3a05f2c5b9a692ea66f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.965ex; height:6.176ex;" alt="{\displaystyle \mathbf {z} (t)=\int _{0}^{t}e^{-uA}\mathbf {b} (u)\,du+\mathbf {c} }"></span></dd></dl> <p>因此， </p> <dl><dd><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 {\begin{aligned}\mathbf {y} _{p}&{}=e^{tA}\int _{0}^{t}e^{-uA}\mathbf {b} (u)\,du+e^{tA}\mathbf {c} \\&{}=\int _{0}^{t}e^{(t-u)A}\mathbf {b} (u)\,du+e^{tA}\mathbf {c} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>u</mi> <mi>A</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>A</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>A</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {y} _{p}&{}=e^{tA}\int _{0}^{t}e^{-uA}\mathbf {b} (u)\,du+e^{tA}\mathbf {c} \\&{}=\int _{0}^{t}e^{(t-u)A}\mathbf {b} (u)\,du+e^{tA}\mathbf {c} \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/622ff239585eff692f8cbba8fafb78ed5e7c589c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:33.021ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {y} _{p}&{}=e^{tA}\int _{0}^{t}e^{-uA}\mathbf {b} (u)\,du+e^{tA}\mathbf {c} \\&{}=\int _{0}^{t}e^{(t-u)A}\mathbf {b} (u)\,du+e^{tA}\mathbf {c} \end{aligned}}}"></span></dd></dl> <p>其中<i><b>c</b></i>由问题的初始条件决定。 </p> <div class="mw-heading mw-heading4"><h4 id="例子（非齐次）"><span id=".E4.BE.8B.E5.AD.90.EF.BC.88.E9.9D.9E.E9.BD.90.E6.AC.A1.EF.BC.89"></span>例子（非齐次）</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=18" title="编辑章节：例子（非齐次）"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>假设我们有以下的微分方程组： </p> <dl><dd><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 {\begin{matrix}x'&=&2x&-y&+z&+e^{2t}\\y'&=&&3y&-1z&\\z'&=&2x&+y&+3z&+e^{2t}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>x</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mo>−</mo> <mi>y</mi> </mtd> <mtd> <mo>+</mo> <mi>z</mi> </mtd> <mtd> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <msup> <mi>y</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd /> <mtd> <mn>3</mn> <mi>y</mi> </mtd> <mtd> <mo>−</mo> <mn>1</mn> <mi>z</mi> </mtd> <mtd /> </mtr> <mtr> <mtd> <msup> <mi>z</mi> <mo>′</mo> </msup> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>2</mn> <mi>x</mi> </mtd> <mtd> <mo>+</mo> <mi>y</mi> </mtd> <mtd> <mo>+</mo> <mn>3</mn> <mi>z</mi> </mtd> <mtd> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}x'&=&2x&-y&+z&+e^{2t}\\y'&=&&3y&-1z&\\z'&=&2x&+y&+3z&+e^{2t}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/965c3f4e4aedd0b1a896fc04e4b1457730c94f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:30.241ex; height:9.509ex;" alt="{\displaystyle {\begin{matrix}x'&=&2x&-y&+z&+e^{2t}\\y'&=&&3y&-1z&\\z'&=&2x&+y&+3z&+e^{2t}\end{matrix}}}"></span></dd></dl> <p>那么我们有 </p> <dl><dd><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 M={\begin{bmatrix}2&-1&1\\0&3&-1\\2&1&3\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M={\begin{bmatrix}2&-1&1\\0&3&-1\\2&1&3\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e69a49c6ba7d4a025867a570b62b3d826b4cc3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:21.142ex; height:9.176ex;" alt="{\displaystyle M={\begin{bmatrix}2&-1&1\\0&3&-1\\2&1&3\end{bmatrix}}}"></span></dd></dl> <p>以及 </p> <dl><dd><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 \mathbf {b} =e^{2t}{\begin{bmatrix}1\\0\\1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">b</mi> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {b} =e^{2t}{\begin{bmatrix}1\\0\\1\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78e41cee797210bbb03ae8598ee1a40d373d02f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:12.33ex; height:9.176ex;" alt="{\displaystyle \mathbf {b} =e^{2t}{\begin{bmatrix}1\\0\\1\end{bmatrix}}}"></span></dd></dl> <p>用前面的方法，我们可以得出齐次微分方程的解。由于齐次方程的通解与非齐次方程的特解的和就是非齐次方程的通解，因此我们只需要找到一个特解（用参数变换法）。 </p><p>我们有： </p> <dl><dd><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 \mathbf {y} _{p}=e^{t}\int _{0}^{t}e^{(-u)M}{\begin{bmatrix}e^{2u}\\0\\e^{2u}\end{bmatrix}}\,du+e^{tM}\mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mi>M</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>M</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}=e^{t}\int _{0}^{t}e^{(-u)M}{\begin{bmatrix}e^{2u}\\0\\e^{2u}\end{bmatrix}}\,du+e^{tM}\mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25522b2ac4aacd1583d8d91573596a6137795779" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:35.611ex; height:9.509ex;" alt="{\displaystyle \mathbf {y} _{p}=e^{t}\int _{0}^{t}e^{(-u)M}{\begin{bmatrix}e^{2u}\\0\\e^{2u}\end{bmatrix}}\,du+e^{tM}\mathbf {c} }"></span></dd></dl> <dl><dd><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 \mathbf {y} _{p}=e^{t}\int _{0}^{t}{\begin{bmatrix}2e^{u}-2ue^{2u}&-2ue^{2u}&0\\\\-2e^{u}+2(u+1)e^{2u}&2(u+1)e^{2u}&0\\\\2ue^{2u}&2ue^{2u}&2e^{u}\end{bmatrix}}{\begin{bmatrix}e^{2u}\\0\\e^{2u}\end{bmatrix}}\,du+e^{tM}\mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>u</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>u</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mo stretchy="false">(</mo> <mi>u</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>2</mn> <mi>u</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mi>u</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>M</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}=e^{t}\int _{0}^{t}{\begin{bmatrix}2e^{u}-2ue^{2u}&-2ue^{2u}&0\\\\-2e^{u}+2(u+1)e^{2u}&2(u+1)e^{2u}&0\\\\2ue^{2u}&2ue^{2u}&2e^{u}\end{bmatrix}}{\begin{bmatrix}e^{2u}\\0\\e^{2u}\end{bmatrix}}\,du+e^{tM}\mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d93e801b0f1d07d014904a2a9ca4e4ba71385d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; margin-top: -0.247ex; width:71.817ex; height:16.176ex;" alt="{\displaystyle \mathbf {y} _{p}=e^{t}\int _{0}^{t}{\begin{bmatrix}2e^{u}-2ue^{2u}&-2ue^{2u}&0\\\\-2e^{u}+2(u+1)e^{2u}&2(u+1)e^{2u}&0\\\\2ue^{2u}&2ue^{2u}&2e^{u}\end{bmatrix}}{\begin{bmatrix}e^{2u}\\0\\e^{2u}\end{bmatrix}}\,du+e^{tM}\mathbf {c} }"></span></dd></dl> <dl><dd><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 \mathbf {y} _{p}=e^{t}\int _{0}^{t}{\begin{bmatrix}e^{2u}(2e^{u}-2ue^{2u})\\\\e^{2u}(-2e^{u}+2(1+u)e^{2u})\\\\2e^{3u}+2ue^{4u}\end{bmatrix}}+e^{tM}\mathbf {c} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>u</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>u</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>u</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>u</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mi>u</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> <mi>u</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>M</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">c</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}=e^{t}\int _{0}^{t}{\begin{bmatrix}e^{2u}(2e^{u}-2ue^{2u})\\\\e^{2u}(-2e^{u}+2(1+u)e^{2u})\\\\2e^{3u}+2ue^{4u}\end{bmatrix}}+e^{tM}\mathbf {c} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76cb634145ad16a266d98f7b41b652c057195b9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.505ex; margin-top: -0.178ex; width:47.396ex; height:16.176ex;" alt="{\displaystyle \mathbf {y} _{p}=e^{t}\int _{0}^{t}{\begin{bmatrix}e^{2u}(2e^{u}-2ue^{2u})\\\\e^{2u}(-2e^{u}+2(1+u)e^{2u})\\\\2e^{3u}+2ue^{4u}\end{bmatrix}}+e^{tM}\mathbf {c} }"></span></dd></dl> <dl><dd><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 \mathbf {y} _{p}=e^{t}{\begin{bmatrix}-{1 \over 24}e^{3t}(3e^{t}(4t-1)-16)\\\\{1 \over 24}e^{3t}(3e^{t}(4t+4)-16)\\\\{1 \over 24}e^{3t}(3e^{t}(4t-1)-16)\end{bmatrix}}+{\begin{bmatrix}2e^{t}-2te^{2t}&-2te^{2t}&0\\\\-2e^{t}+2(t+1)e^{2t}&2(t+1)e^{2t}&0\\\\2te^{2t}&2te^{2t}&2e^{t}\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo>+</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>24</mn> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mn>4</mn> <mi>t</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mn>16</mn> <mo stretchy="false">)</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mo>−</mo> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> <mo>+</mo> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd /> </mtr> <mtr> <mtd> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <mi>t</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>t</mi> </mrow> </msup> </mtd> <mtd> <mn>2</mn> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} _{p}=e^{t}{\begin{bmatrix}-{1 \over 24}e^{3t}(3e^{t}(4t-1)-16)\\\\{1 \over 24}e^{3t}(3e^{t}(4t+4)-16)\\\\{1 \over 24}e^{3t}(3e^{t}(4t-1)-16)\end{bmatrix}}+{\begin{bmatrix}2e^{t}-2te^{2t}&-2te^{2t}&0\\\\-2e^{t}+2(t+1)e^{2t}&2(t+1)e^{2t}&0\\\\2te^{2t}&2te^{2t}&2e^{t}\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5a5abd6fc2c52dc5fb0cb0ce12f707d6e90eb85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.274ex; margin-bottom: -0.231ex; width:85.345ex; height:18.176ex;" alt="{\displaystyle \mathbf {y} _{p}=e^{t}{\begin{bmatrix}-{1 \over 24}e^{3t}(3e^{t}(4t-1)-16)\\\\{1 \over 24}e^{3t}(3e^{t}(4t+4)-16)\\\\{1 \over 24}e^{3t}(3e^{t}(4t-1)-16)\end{bmatrix}}+{\begin{bmatrix}2e^{t}-2te^{2t}&-2te^{2t}&0\\\\-2e^{t}+2(t+1)e^{2t}&2(t+1)e^{2t}&0\\\\2te^{2t}&2te^{2t}&2e^{t}\end{bmatrix}}{\begin{bmatrix}c_{1}\\c_{2}\\c_{3}\end{bmatrix}}}"></span></dd></dl> <p>进一步简化，就可以得到原方程的特解。 </p> <div class="mw-heading mw-heading2"><h2 id="註釋"><span id=".E8.A8.BB.E9.87.8B"></span>註釋</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=19" title="编辑章节：註釋"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="reflist" style="list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a><span class="error harv-error" style="display: none; font-size:100%"> harvnb模板錯誤: 無指向目標: CITEREFHall2015 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%93%88%E4%BD%9B%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE%E6%A0%BC%E5%BC%8F%E7%B3%BB%E5%88%97%E6%A8%A1%E6%9D%BF%E9%93%BE%E6%8E%A5%E6%8C%87%E5%90%91%E9%94%99%E8%AF%AF%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:含有哈佛参考文献格式系列模板链接指向错误的页面">幫助</a>)</span> Proposition 2.3</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a><span class="error harv-error" style="display: none; font-size:100%"> harvnb模板錯誤: 無指向目標: CITEREFHall2015 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%93%88%E4%BD%9B%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE%E6%A0%BC%E5%BC%8F%E7%B3%BB%E5%88%97%E6%A8%A1%E6%9D%BF%E9%93%BE%E6%8E%A5%E6%8C%87%E5%90%91%E9%94%99%E8%AF%AF%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:含有哈佛参考文献格式系列模板链接指向错误的页面">幫助</a>)</span> Theorem 2.12</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">(<a href="#CITEREFHornJohnson1991">Horn & Johnson 1991</a>，第435–437頁)</span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFHall2015">Hall 2015</a><span class="error harv-error" style="display: none; font-size:100%"> harvnb模板錯誤: 無指向目標: CITEREFHall2015 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%93%88%E4%BD%9B%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE%E6%A0%BC%E5%BC%8F%E7%B3%BB%E5%88%97%E6%A8%A1%E6%9D%BF%E9%93%BE%E6%8E%A5%E6%8C%87%E5%90%91%E9%94%99%E8%AF%AF%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:含有哈佛参考文献格式系列模板链接指向错误的页面">幫助</a>)</span> Theorem 2.11</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"><a href="#CITEREFHall2015">Hall 2015</a><span class="error harv-error" style="display: none; font-size:100%"> harvnb模板錯誤: 無指向目標: CITEREFHall2015 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%93%88%E4%BD%9B%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE%E6%A0%BC%E5%BC%8F%E7%B3%BB%E5%88%97%E6%A8%A1%E6%9D%BF%E9%93%BE%E6%8E%A5%E6%8C%87%E5%90%91%E9%94%99%E8%AF%AF%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:含有哈佛参考文献格式系列模板链接指向错误的页面">幫助</a>)</span> Chapter 5</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"><cite class="citation web"><a rel="nofollow" class="external text" href="http://www.mathworks.de/help/techdoc/ref/expm.html">矩阵指数 - MATLAB expm - MathWorks Deutschland</a>. Mathworks. 2011-04-30 <span class="reference-accessdate"> [<span class="nowrap">2013-06-05</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20120730224156/http://www.mathworks.de/help/techdoc/ref/expm.html">存档</a>于2012-07-30）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&rft.btitle=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0+-+MATLAB+expm+-+MathWorks+Deutschland&rft.date=2011-04-30&rft.genre=unknown&rft.pub=Mathworks.&rft_id=http%3A%2F%2Fwww.mathworks.de%2Fhelp%2Ftechdoc%2Fref%2Fexpm.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"><cite class="citation web"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20150529200317/http://www.network-theory.co.uk/docs/octave3/octave_200.html">GNU Octave - 矩阵的函数</a>. Network-theory.co.uk. 2007-01-11 <span class="reference-accessdate"> [<span class="nowrap">2013-06-05</span>]</span>. （<a rel="nofollow" class="external text" href="http://www.network-theory.co.uk/docs/octave3/octave_200.html">原始内容</a>存档于2015-05-29）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&rft.btitle=GNU+Octave+-+%E7%9F%A9%E9%98%B5%E7%9A%84%E5%87%BD%E6%95%B0&rft.date=2007-01-11&rft.genre=unknown&rft.pub=Network-theory.co.uk&rft_id=http%3A%2F%2Fwww.network-theory.co.uk%2Fdocs%2Foctave3%2Foctave_200.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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"><cite class="citation web"><a rel="nofollow" class="external text" href="http://docs.scipy.org/doc/scipy-0.15.1/reference/generated/scipy.linalg.expm.html">scipy.linalg. expm函数文档</a>. The SciPy Community. 2015-01-18 <span class="reference-accessdate"> [<span class="nowrap">2015-05-29</span>]</span>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20210207182321/https://docs.scipy.org/doc/scipy-0.15.1/reference/generated/scipy.linalg.expm.html">存档</a>于2021-02-07）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&rft.btitle=scipy.linalg.+expm%E5%87%BD%E6%95%B0%E6%96%87%E6%A1%A3&rft.date=2015-01-18&rft.genre=unknown&rft.pub=The+SciPy+Community&rft_id=http%3A%2F%2Fdocs.scipy.org%2Fdoc%2Fscipy-0.15.1%2Freference%2Fgenerated%2Fscipy.linalg.expm.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></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">见<a href="#CITEREFHall2015">Hall 2015</a><span class="error harv-error" style="display: none; font-size:100%"> harvnb模板錯誤: 無指向目標: CITEREFHall2015 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%93%88%E4%BD%9B%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE%E6%A0%BC%E5%BC%8F%E7%B3%BB%E5%88%97%E6%A8%A1%E6%9D%BF%E9%93%BE%E6%8E%A5%E6%8C%87%E5%90%91%E9%94%99%E8%AF%AF%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:含有哈佛参考文献格式系列模板链接指向错误的页面">幫助</a>)</span>。2.2节</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=20" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r80540462">.mw-parser-output .refbegin{font-size:90%;margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .refbegin-100{font-size:100%}</style><div class="refbegin" style=""> <ul><li><cite id="CITEREFHornJohnson1991" class="citation">Horn, Roger A.; Johnson, Charles R., Topics in Matrix Analysis, <a href="/wiki/Cambridge_University_Press" class="mw-redirect" title="Cambridge University Press">Cambridge University Press</a>, 1991, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-521-46713-1" title="Special:网络书源/978-0-521-46713-1"><span title="国际标准书号">ISBN</span> 978-0-521-46713-1</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&rft.au=Johnson%2C+Charles+R.&rft.aufirst=Roger+A.&rft.aulast=Horn&rft.btitle=Topics+in+Matrix+Analysis&rft.date=1991&rft.genre=book&rft.isbn=978-0-521-46713-1&rft.pub=Cambridge+University+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span>.</li> <li><cite id="CITEREFMolerVan_Loan2003" class="citation"><a href="/w/index.php?title=Cleve_Moler&action=edit&redlink=1" class="new" title="Cleve Moler（页面不存在）">Moler, Cleve</a>; <a href="/w/index.php?title=Charles_F._Van_Loan&action=edit&redlink=1" class="new" title="Charles F. Van Loan（页面不存在）">Van Loan, Charles F.</a>, <a rel="nofollow" class="external text" href="http://www.cs.cornell.edu/cv/researchpdf/19ways+.pdf">Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later</a> <span style="font-size:85%;">(PDF)</span>, SIAM Review, 2003, <b>45</b> (1): 3–49 <span class="reference-accessdate"> [<span class="nowrap">2008-08-14</span>]</span>, <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1095-7200"><span title="国际标准连续出版物号">ISSN 1095-7200</span></a>, （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20081208020310/http://www.cs.cornell.edu/cv/ResearchPDF/19ways+.pdf">存档</a> <span style="font-size:85%;">(PDF)</span>于2008-12-08）</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&rft.atitle=Nineteen+Dubious+Ways+to+Compute+the+Exponential+of+a+Matrix%2C+Twenty-Five+Years+Later&rft.au=Van+Loan%2C+Charles+F.&rft.aufirst=Cleve&rft.aulast=Moler&rft.date=2003&rft.genre=article&rft.issn=1095-7200&rft.issue=1&rft.jtitle=SIAM+Review&rft.pages=3-49&rft.volume=45&rft_id=http%3A%2F%2Fwww.cs.cornell.edu%2Fcv%2Fresearchpdf%2F19ways%2B.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span>.</li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="参閱"><span id=".E5.8F.82.E9.96.B1"></span>参閱</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=21" title="编辑章节：参閱"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r79074265">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style><div class="div-col" style="column-count:2; column-width:auto;"> <ul><li><a href="/wiki/%E7%9F%A9%E9%98%B5%E5%AF%B9%E6%95%B0" title="矩阵对数">矩阵对数</a></li> <li><a href="/wiki/%E6%8C%87%E6%95%B0%E5%87%BD%E6%95%B0" title="指数函数">指数函数</a></li> <li><a href="/wiki/%E6%8C%87%E6%95%B8%E6%98%A0%E5%B0%84_(%E6%9D%8E%E7%BE%A4)" title="指數映射 (李群)">指数映射</a></li> <li><a href="/w/index.php?title=%E5%90%91%E9%87%8F%E6%B5%81&action=edit&redlink=1" class="new" title="向量流（页面不存在）">向量流</a></li> <li><a href="/wiki/%E9%AB%98%E7%99%BB%E2%94%80%E6%B9%AF%E6%99%AE%E6%A3%AE%E4%B8%8D%E7%AD%89%E5%BC%8F" class="mw-redirect" title="高登─湯普森不等式">高登─湯普森不等式</a></li> <li><a href="/w/index.php?title=%E4%BD%8D%E7%9B%B8%E5%9E%8B%E5%88%86%E5%B8%83&action=edit&redlink=1" class="new" title="位相型分布（页面不存在）">位相型分布</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="外部链接"><span id=".E5.A4.96.E9.83.A8.E9.93.BE.E6.8E.A5"></span>外部链接</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&action=edit&section=22" title="编辑章节：外部链接"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="citation mathworld" id="Reference-Mathworld-Matrix_Exponential"><cite class="citation web"><a href="/wiki/%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0" title="埃里克·韦斯坦因">埃里克·韦斯坦因</a>. <a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/MatrixExponential.html">Matrix Exponential</a>. <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&rft.atitle=Matrix+Exponential&rft.au=%E5%9F%83%E9%87%8C%E5%85%8B%C2%B7%E9%9F%A6%E6%96%AF%E5%9D%A6%E5%9B%A0&rft.genre=unknown&rft.jtitle=MathWorld&rft_id=http%3A%2F%2Fmathworld.wolfram.com%2FMatrixExponential.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"><span style="display:none;"> </span></span></span></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20070609200917/http://math.fullerton.edu/mathews/n2003/MatrixExponentialMod.html">矩阵指数的教程</a></li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=矩阵指数&oldid=84499588">https://zh.wikipedia.org/w/index.php?title=矩阵指数&oldid=84499588</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" title="Special:页面分类">分类</a>：<ul><li><a href="/wiki/Category:%E7%9F%A9%E9%99%A3%E8%AB%96" title="Category:矩陣論">矩陣論</a></li><li><a href="/wiki/Category:%E6%9D%8E%E7%BE%A4" title="Category:李群">李群</a></li><li><a href="/wiki/Category:%E6%8C%87%E6%95%B0" title="Category:指数">指数</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">隐藏分类：<ul><li><a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%93%88%E4%BD%9B%E5%8F%82%E8%80%83%E6%96%87%E7%8C%AE%E6%A0%BC%E5%BC%8F%E7%B3%BB%E5%88%97%E6%A8%A1%E6%9D%BF%E9%93%BE%E6%8E%A5%E6%8C%87%E5%90%91%E9%94%99%E8%AF%AF%E7%9A%84%E9%A1%B5%E9%9D%A2" title="Category:含有哈佛参考文献格式系列模板链接指向错误的页面">含有哈佛参考文献格式系列模板链接指向错误的页面</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"> 本页面最后修订于2024年10月7日 (星期一) 15:01。</li> <li id="footer-info-copyright">本站的全部文字在<a rel="nofollow" class="external text" href="//creativecommons.org/licenses/by-sa/4.0/deed.zh">知识共享署名-相同方式共享 4.0协议</a>之条款下提供，附加条款亦可能应用。（请参阅<a class="external text" href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use">使用条款</a>）<br /> Wikipedia®和维基百科标志是<a rel="nofollow" class="external text" href="https://wikimediafoundation.org/zh">维基媒体基金会</a>的注册商标；维基™是维基媒体基金会的商标。<br /> 维基媒体基金会是按美国国內稅收法501(c)(3)登记的<a class="external text" href="https://donate.wikimedia.org/wiki/Special:MyLanguage/Tax_deductibility">非营利慈善机构</a>。<br /></li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">隐私政策</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:%E5%85%B3%E4%BA%8E">关于维基百科</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:%E5%85%8D%E8%B4%A3%E5%A3%B0%E6%98%8E">免责声明</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">行为准则</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">开发者</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/zh.wikipedia.org">统计</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie声明</a></li> <li id="footer-places-mobileview"><a href="//zh.m.wikipedia.org/w/index.php?title=%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">手机版视图</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"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></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"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </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.main-5c59558b9d-sf7pf","wgBackendResponseTime":178,"wgPageParseReport":{"limitreport":{"cputime":"0.269","walltime":"0.442","ppvisitednodes":{"value":1184,"limit":1000000},"postexpandincludesize":{"value":19180,"limit":2097152},"templateargumentsize":{"value":897,"limit":2097152},"expansiondepth":{"value":6,"limit":100},"expensivefunctioncount":{"value":0,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":11807,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 194.670 1 -total"," 57.21% 111.370 1 Template:Reflist"," 31.62% 61.556 4 Template:Cite_web"," 20.79% 40.467 5 Template:Harvnb"," 7.80% 15.190 1 Template:Equation_box_1"," 7.28% 14.166 1 Template:Refbegin"," 6.40% 12.463 2 Template:Citation"," 5.10% 9.927 7 Template:Math"," 4.56% 8.885 1 Template:Div_col"," 3.67% 7.135 7 Template:Serif"]},"scribunto":{"limitreport-timeusage":{"value":"0.069","limit":"10.000"},"limitreport-memusage":{"value":3469082,"limit":52428800},"limitreport-logs":"anchor_id_list = table#1 {\n [\"CITEREFHornJohnson1991\"] = 1,\n [\"CITEREFMolerVan_Loan2003\"] = 1,\n}\ntemplate_list = table#1 {\n [\"=\"] = 4,\n [\"Citation\"] = 2,\n [\"Cite web\"] = 3,\n [\"Div col\"] = 1,\n [\"Div col end\"] = 1,\n [\"Equation box 1\"] = 1,\n [\"Harv\"] = 1,\n [\"Harvnb\"] = 5,\n [\"Math\"] = 7,\n [\"Mathworld\"] = 1,\n [\"Refbegin\"] = 1,\n [\"Refend\"] = 1,\n [\"Reflist\"] = 1,\n}\narticle_whitelist = table#1 {\n}\n"},"cachereport":{"origin":"mw-api-int.codfw.main-ccb4d9f9f-mx2b5","timestamp":"20241107190833","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"\u77e9\u9635\u6307\u6570","url":"https:\/\/zh.wikipedia.org\/wiki\/%E7%9F%A9%E9%98%B5%E6%8C%87%E6%95%B0","sameAs":"http:\/\/www.wikidata.org\/entity\/Q1191722","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q1191722","author":{"@type":"Organization","name":"\u7ef4\u57fa\u5a92\u4f53\u9879\u76ee\u8d21\u732e\u8005"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2008-08-14T17:43:57Z","dateModified":"2024-10-07T15:01:02Z"}</script> </body> </html>

CINXE.COM

矩阵指数 - 维基百科，自由的百科全书