{"id":8284,"date":"2022-11-30T13:18:01","date_gmt":"2022-11-30T05:18:01","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=8284"},"modified":"2025-09-07T22:05:31","modified_gmt":"2025-09-07T14:05:31","slug":"%e7%9f%a9%e9%98%b5%e5%b8%b8%e7%94%a8%e6%80%a7%e8%b4%a8","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=8284","title":{"rendered":"\u77e9\u9635\u6027\u8d28"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/blog.iyatt.com\/?p=8284\/#n_%E9%98%B6%E7%9F%A9%E9%98%B5%E7%9A%84%E8%A1%8C%E5%88%97%E5%BC%8F%E6%80%A7%E8%B4%A8\" >n \u9636\u77e9\u9635\u7684\u884c\u5217\u5f0f\u6027\u8d28<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/blog.iyatt.com\/?p=8284\/#%E7%9F%A9%E9%98%B5%E8%BF%90%E7%AE%97%E6%80%A7%E8%B4%A8\" >\u77e9\u9635\u8fd0\u7b97\u6027\u8d28<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/blog.iyatt.com\/?p=8284\/#%E7%9F%A9%E9%98%B5%E8%BD%AC%E7%BD%AE%E6%80%A7%E8%B4%A8\" >\u77e9\u9635\u8f6c\u7f6e\u6027\u8d28<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/blog.iyatt.com\/?p=8284\/#%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5%E6%80%A7%E8%B4%A8\" >\u4f34\u968f\u77e9\u9635\u6027\u8d28<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/blog.iyatt.com\/?p=8284\/#%E9%80%86%E7%9F%A9%E9%98%B5%E6%80%A7%E8%B4%A8\" >\u9006\u77e9\u9635\u6027\u8d28<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/blog.iyatt.com\/?p=8284\/#%E7%9F%A9%E9%98%B5%E7%A7%A9%E7%9A%84%E6%80%A7%E8%B4%A8\" >\u77e9\u9635\u79e9\u7684\u6027\u8d28<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"n_%E9%98%B6%E7%9F%A9%E9%98%B5%E7%9A%84%E8%A1%8C%E5%88%97%E5%BC%8F%E6%80%A7%E8%B4%A8\"><\/span>n \u9636\u77e9\u9635\u7684\u884c\u5217\u5f0f\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u8bbe A\u3001B \u4e3a n \u9636\u77e9\u9635<\/p>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n|A^T|=|A|\\\\\n|A^m|=|A|^m\\\\\n|kA|=k^n|A|\\\\\n|AB|=|A||B|\\\\\n|A^{-1}|=\\frac1{|A|}\\\\\n|A^*|=|A|^{n-1}\n\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E7%9F%A9%E9%98%B5%E8%BF%90%E7%AE%97%E6%80%A7%E8%B4%A8\"><\/span>\u77e9\u9635\u8fd0\u7b97\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nA+B=B+A \\\\\n(A+B)+C=A+(B+C) \\\\\n(AB)C=A(BC) \\\\\n(k+l)A=kA+lA \\\\\nk(A+B)=kA+kB \\\\\nA(B+C)=AB+AC \\\\ \\\\\n\n\u8bbe A \u662f\\ m\\times n\\ \u77e9\u9635\uff0cB\u3001C\\ \u5206\u522b\u662f\\ n\\times s\u3001n\\times l\\ \u77e9\u9635\uff0c\u5219 \\\\\nA(B\\vdots C)=(AB\\vdots AC)\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E7%9F%A9%E9%98%B5%E8%BD%AC%E7%BD%AE%E6%80%A7%E8%B4%A8\"><\/span>\u77e9\u9635\u8f6c\u7f6e\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n(A^T)^T=A \\\\\n(kA)^T=kA^T \\\\\n(A\\pm B)^T=A^T\\pm B^T \\\\\n(AB)^T=B^TA^T \\\\\n(A^{-1})^T=(A^T)^{-1} \\\\\n(A^m)^T=(A^T)^m\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5%E6%80%A7%E8%B4%A8\"><\/span>\u4f34\u968f\u77e9\u9635\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n(A^T)^*=(A^*)^T \\\\\n(kA)^*=k^{n-1}A^* \\\\\n(AB)^*=B^*A^*\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E9%80%86%E7%9F%A9%E9%98%B5%E6%80%A7%E8%B4%A8\"><\/span>\u9006\u77e9\u9635\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n(A^{-1})^{-1}=A \\\\\n(kA)^{-1}=\\frac{1}{k}A^{-1} \\\\\n(AB)^{-1}=B^{-1}A^{-1} \\\\\n(A^T)^{-1}=(A^{-1})^T \\\\\n(A^n)^{-1}=(A^{-1})^n \\\\ \\\\\n\n\u8bbe\\ A\u3001B\\ \u5206\u522b\u4e3a\\ m\\ \u548c\\ n\\ \u9636\u77e9\u9635\uff0c\u5219 \\\\\n\\left (\n\\begin{matrix}\nA &amp;O \\\\\nO &amp;B\n\\end{matrix}\n\\right )\n^{-1}=\n\\left (\n\\begin{matrix}\nA^{-1} &amp;O \\\\\nO &amp;B^{-1}\n\\end{matrix}\n\\right ) \\\\\n\n\\left (\n\\begin{matrix}\nO &amp;A \\\\\nB &amp;O\n\\end{matrix}\n\\right )\n^{-1}=\n\\left (\n\\begin{matrix}\nO &amp;B^{-1} \\\\\nA^{-1} &amp;O\n\\end{matrix}\n\\right )\n\\\\ \\\\\n\u5bf9\u4e8e n \u9636\u53ef\u9006\u77e9\u9635 A\uff0cA^{-1}=\\frac1{|A|}A^*\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E7%9F%A9%E9%98%B5%E7%A7%A9%E7%9A%84%E6%80%A7%E8%B4%A8\"><\/span>\u77e9\u9635\u79e9\u7684\u6027\u8d28<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{array}{l}\nr(A)=r(A^T)=r(A^TA)=r(AA^T) \\\\ \\\\\n\n\u8bbe\\ A\uff0cB\\ \u662f\u540c\u578b\u77e9\u9635\uff0c\u5219\\ r(A\\pm B)\\le r(A)+r(B) \\\\ \\\\\n\n\u8bbe\\ A \u4e3a\\ m\\times n\\ \u77e9\u9635\uff0cB\\ \u4e3a\\ n\\times s\\ \u77e9\u9635\uff0c\u5219\\ r(AB)\\le min\\{r(A),r(B)\\} \\\\ \\\\\n\n\u8bbe\\ A\uff0cB\\ \u5206\u522b\u4e3a\\ m\\times n\\ \u53ca\\ n\\times s\\ \u77e9\u9635\uff0c\u4e14 AB=O\uff0c\u5219 r(A)+r(B)\\le n \\\\ \\\\\n\n\u8bbe\\ A\\ \u662f\\ m\\times n\\ \u9636\u77e9\u9635\uff0cP\uff0cQ\\ \u5206\u522b\u4e3a\\ m \u53ca\\ n\\ \u9636\u77e9\u9635\uff0c\u5219 \\\\\nr(A)=r(PA)=r(AQ)=r(PAQ) \\\\ \\\\\n\n\u8bbe\\ A\\ \u662f\\ n\\ \u9636\u77e9\u9635\uff0c\u5219\\ r(A^*)= \\\\\n\\begin{cases}\nn,\\ r(A)=n \\\\\n1,\\ r(A)=n-1,\\ (n\\ge2) \\\\\n0,\\ r(A)\\lt n-1\n\\end{cases} \\\\ \\\\\n\n\u8bbe\\ A\u3001B\\ \u5206\u522b\u4e3a\\ m\\times s\u3001n\\times s\\ \u9636\u77e9\u9635\uff0c\u5219 \\\\\nmax\\{r(A),r(B)\\}\\le r\n\\left (\n\\begin{matrix}\nA \\\\\nB\n\\end{matrix}\n\\right )\n\\le r(A)+r(B) \\\\\n\u6216\u8bbe\\ A\u3001B\\ \u5206\u522b\u4e3a\\ m\\times n\u3001m\\times s\\ \u77e9\u9635\uff0c\u5219 \\\\\nmax\\{r(A),r(B)\\}\\le r(A\\vdots B)\\le r(A)+r(B)\u3002\\\\\nr\n\\left (\n\\begin{matrix}\nA &amp;O \\\\\nO &amp;B\n\\end{matrix}\n\\right )\n=r(A)+r(B) \\\\ \\\\\n\n\u8bbe\\ A\\ \u4e3a\\ n\\ \u9636\u975e\u96f6\u77e9\u9635\uff0c\u5219\\ r(A)=1\\ \u7684\u5145\u5206\u5fc5\u8981\u6761\u4ef6\u662f\uff0c\\\\\n\\exists\\ \u975e\u96f6\u5411\u91cf\\ \\alpha,\\beta, \u4f7f\u5f97\\ A=\\alpha\\beta^T\n\n\\end{array}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>n \u9636\u77e9\u9635\u7684\u884c\u5217\u5f0f\u6027\u8d28 \u8bbe A\u3001B \u4e3a n \u9636\u77e9\u9635 \\begin{array}{l} |A^T|=|A|\\\\  [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,612],"tags":[1648,1652,1653,1649,1647,1657,1655,1656,1650,945,1658,944,1630,1651,1654],"class_list":["post-8284","post","type-post","status-publish","format-standard","hentry","category-all","category-612","tag-n-","tag-1652","tag-1653","tag-1649","tag-1647","tag-1657","tag-1655","tag-1656","tag-1650","tag-945","tag-1658","tag-944","tag-1630","tag-1651","tag-1654"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/8284","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=8284"}],"version-history":[{"count":8,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/8284\/revisions"}],"predecessor-version":[{"id":21566,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/8284\/revisions\/21566"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=8284"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=8284"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=8284"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}