{"id":252,"date":"2021-04-09T16:51:38","date_gmt":"2021-04-09T08:51:38","guid":{"rendered":"http:\/\/www.wayln.com\/?p=252"},"modified":"2021-04-15T10:51:21","modified_gmt":"2021-04-15T02:51:21","slug":"%e7%ac%ac%e4%ba%8c%e7%ab%a0-%e7%9f%a9%e9%98%b5%e4%bb%a3%e6%95%b0","status":"publish","type":"post","link":"http:\/\/www.wayln.com\/?p=252","title":{"rendered":"\u7b2c\u4e8c\u7ae0 \u77e9\u9635\u4ee3\u6570"},"content":{"rendered":"<div id=\"toc_container\" class=\"toc_transparent no_bullets\"><p class=\"toc_title\">Contents<\/p><ul class=\"toc_list\"><li><a href=\"#21\"><span class=\"toc_number toc_depth_1\">1<\/span> 2.1 \u77e9\u9635\u8fd0\u7b97<\/a><ul><li><a href=\"#i\"><span class=\"toc_number toc_depth_2\">1.1<\/span> \u548c\u4e0e\u6807\u91cf\u4e58\u6cd5<\/a><\/li><li><a href=\"#i-2\"><span class=\"toc_number toc_depth_2\">1.2<\/span> \u77e9\u9635\u4e58\u6cd5<\/a><\/li><li><a href=\"#i-3\"><span class=\"toc_number toc_depth_2\">1.3<\/span> \u77e9\u9635\u4e58\u6cd5\u7684\u6027\u8d28<\/a><\/li><li><a href=\"#i-4\"><span class=\"toc_number toc_depth_2\">1.4<\/span> \u77e9\u9635\u7684\u4e58\u5e42<\/a><\/li><li><a href=\"#i-5\"><span class=\"toc_number toc_depth_2\">1.5<\/span> \u77e9\u9635\u7684\u8f6c\u7f6e<\/a><\/li><\/ul><\/li><\/ul><\/div>\n\n<h1><span id=\"21\">2.1 \u77e9\u9635\u8fd0\u7b97<\/span><\/h1>\n<p>\u82e5A\u662fmxn\u77e9\u9635\uff0c\u5373\u7531m\u884cn\u5217\u7684\u77e9\u9635\uff0cA\u7684\u7b2ci\u884c\u7b2cj\u5217\u7684\u5143\u7d20\u7528<span class=\"katex math inline\">a_{ij}<\/span>\u8868\u793a\uff0c\u6210\u4e3aA\u7684(i,j)\u5143\u7d20\u3002A\u7684\u5404\u5217\u662f<span class=\"katex math inline\">R^m<\/span>\u7684\u5411\u91cf\u7528\uff08\u9ed1\u4f53\u5b57\u6bcd\uff09<span class=\"katex math inline\">a_1,\\dots,a_n<\/span>\u8868\u793a\u3002\u5f53\u6211\u4eec\u7279\u522b\u6ce8\u610fA\u7684\u5404\u5217\u65f6\uff0c\u6211\u4eec\u5199\u6210<span class=\"katex math inline\">A=\\begin{bmatrix}a_1&amp;a_2&amp;\\dots&amp;a_n\\end{bmatrix}<\/span>,\u6ce8\u610f<span class=\"katex math inline\">a_{ij}<\/span>\u662f\u7b2cj\u4e2a\u5217\u5411\u91cf<span class=\"katex math inline\">a_j<\/span>\uff09(\u4ece\u4e0a\u9762\u7b97\u8d77)\u7684\u7b2ci\u4e2a\u5143\u7d20\u3002<br \/>\n<span class=\"katex math inline\">A=\\begin{bmatrix}a_{ij}\\end{bmatrix}<\/span>\u7684\u5bf9\u89d2\u5143\u7d20\u4f7f<span class=\"katex math inline\">a_{11},a_{22},a_{33},\\dots,<\/span>\u4ed6\u4eec\u7ec4\u6210A\u7684\u4e3b\u5bf9\u89d2\u7ebf\u3002<strong>\u5bf9\u89d2\u77e9\u9635<\/strong>\u662f\u4e00\u4e2a\u4eff\u771f\uff0c\u5b83\u7684\u975e\u5bf9\u89d2\u7ebf\u5143\u7d20\u5168\u662f0.\u5143\u7d20\u5168\u662f\u96f6\u7684mxn\u7684\u77e9\u9635\u79f0\u4e3a\u96f6\u77e9\u9635\uff0c\u75280\u8868\u793a\uff0c\u53ef\u4ee5\u7528<span class=\"katex math inline\">0_{mxn}<\/span>\u8868\u793a\u3002<\/p>\n<h2><span id=\"i\">\u548c\u4e0e\u6807\u91cf\u4e58\u6cd5<\/span><\/h2>\n<blockquote><p>\n  \u5b9a\u74061 \u8bbeA\uff0cB,C\u662f\u76f8\u540c\u7ef4\u6570\u7684\u77e9\u9635\uff0cr\u4e0es\u4e3a\u6570\uff0c\u5219\u6709<br \/>\n  a. A+B=B+A<br \/>\n  b. (A+B)+C=A+(B+C)<br \/>\n  c. A+0=A<br \/>\n  d. r(A+B)=rA+rB<br \/>\n  e. (r+s)A=rA+sA<br \/>\n  f. r(sA)=(rs)A\n<\/p><\/blockquote>\n<h2><span id=\"i-2\">\u77e9\u9635\u4e58\u6cd5<\/span><\/h2>\n<p>\u5f53\u628a\u77e9\u9635B\u4e58\u4ee5\u5411\u91cfx\uff0c\u5b83\u5c06x\u53d8\u4e3a\u5411\u91cfBx,\u82e5\u8fd9\u5411\u91cf\u53c8\u4e58\u4ee5\u77e9\u9635A\uff0c\u7ed3\u679c\u5f97\u5411\u91cfA(Bx)<br \/>\n<a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_0af7aa813f2dd125e149e52f73eec95f.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_0af7aa813f2dd125e149e52f73eec95f.jpg\" alt=\"\" \/><\/a><\/p>\n<p>\u6211\u4eec\u7684\u76ee\u7684\u662f\u5c06\u6b64\u7b26\u5408\u6620\u5c04\u8868\u793a\u4e3a\u4e58\u4ee5\u4e00\u4e2a\u77e9\u9635\u7684\u53d8\u6362\uff0c\u6b64\u77e9\u9635\u8bb0\u4e3aAB\uff0c\u5373<\/p>\n<div class=\"katex math multi-line no-emojify\">A(Bx)=(AB)x \\tag{1}<\/p>\n<\/div>\n<p><a class=\"wp-editor-md-post-content-link\" href=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_60f750cc95faea2aa1c5bcb684a3a039.jpg\"><img decoding=\"async\" src=\"http:\/\/www.wayln.com\/wp-content\/uploads\/2021\/04\/wp_editor_md_60f750cc95faea2aa1c5bcb684a3a039.jpg\" alt=\"\" \/><\/a><\/p>\n<blockquote><p>\n  <strong>\u5b9a\u4e49<\/strong> \u82e5A\u662fmxn\u77e9\u9635\uff0cB\u662fnxp\u77e9\u9635\uff0cB\u7684\u5217\u662f<span class=\"katex math inline\">b_1,\\dots,b_p<\/span>\uff0c\u5219\u4e58\u79efAB\u662fmxp\u77e9\u9635\uff0c\u5b83\u7684\u5404\u5217\u662f<span class=\"katex math inline\">Ab_1,\\dots.Ab_p<\/span>,\u5373<\/p>\n<div class=\"katex math multi-line no-emojify\">AB=A\\begin{bmatrix}<br \/>\n  b_1&amp;b_2&amp;\\dots&amp;b_p<br \/>\n  \\end{bmatrix}=<br \/>\n  \\begin{bmatrix}<br \/>\n  Ab_1&amp;Ab_2&amp;\\dots&amp;Ab_p<br \/>\n  \\end{bmatrix}\n  <\/div>\n<p>  AB\u7684\u6bcf\u4e00\u5217\u90fd\u662fA\u7684\u5404\u5217\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u4ee5B\u7684\u5bf9\u5e94\u5217\u7684\u5143\u7d20\u4e3a\u6743\u3002\n<\/p><\/blockquote>\n<p>\u7531\u5b9a\u4e49\u77e5\uff0c<strong>AB\u7684\u884c\u6570\u7b49\u4e8eA\u7684\u884c\u6570\uff0c\u5217\u6570\u7b49\u4e8eB\u7684\u5217\u6570<\/strong><\/p>\n<blockquote><p>\n  <strong>\u8ba1\u7b97AB\u7684\u884c\u5217\u6cd5\u5219<\/strong><br \/>\n  \u82e5\u4e58\u79efAB\u6709\u5b9a\u4e49\uff0cAB\u7684\u7b2ci\u884c\u7b2cj\u5217\u7684\u5143\u7d20\u662fA\u7684\u7b2ci\u884c\u4e0eB\u7684\u7b2cj\u5217\u5bf9\u5e94\u5143\u7d20\u4e58\u79ef\u4e4b\u548c\u3002\u82e5<span class=\"katex math inline\">(AB)_{ij}<\/span>\u8868\u793aAB\u7684(i,j)\u5143\u7d20\uff0cA\u4e3amxn\u77e9\u9635\uff0c\u5219\n<\/p><\/blockquote>\n<div class=\"katex math multi-line no-emojify\">(AB)_{ij}=a_{i1}b_{1j}+a_{i2}b_{2j}+\\dots+a_{in}b_{nj}\n<\/div>\n<h2><span id=\"i-3\">\u77e9\u9635\u4e58\u6cd5\u7684\u6027\u8d28<\/span><\/h2>\n<p><span class=\"katex math inline\">I_m<\/span>\u8868\u793amxm\u5355\u4f4d\u77e9\u9635\uff0c\u5bf9<span class=\"katex math inline\">R^m<\/span>\u4e2d\u7684\u4e00\u5207x,<span class=\"katex math inline\">I_mx=x<\/span>\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74062<\/strong> \u8bbeA\u4e3amxn\u77e9\u9635\uff0cB\u3001C\u7684\u7ef4\u6570\u4f7f\u4e0b\u5217\u5404\u5f0f\u7684\u4e58\u79ef\u6709\u5b9a\u4e49\u3002<br \/>\n  a. A(BC)=(AB)C<br \/>\n  b. A(B+C)=AB+AC<br \/>\n  c. (B+C)A=BA+CA<br \/>\n  d. r(AB)=(rA)B+A(rB),<br \/>\n  e. <span class=\"katex math inline\">I_m<\/span>A=A=A<span class=\"katex math inline\">I_n<\/span>\n<\/p><\/blockquote>\n<p>\u4e58\u79ef\u4e2d\u7684\u5de6\u53f3\u987a\u5e8f\u662f\u91cd\u8981\u7684\uff0c\u56e0\u4e3a\u4e00\u822c\u6765\u8bf4AB\u4e0eBA\u5e76\u4e0d\u76f8\u540c\u3002\u8fd9\u5e76\u4e0d\u5947\u602a\uff0c\u56e0AB\u7684\u5217\u65f6A\u7684\u5404\u5217\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u800cBA\u7684\u5404\u5217\u662fB\u7684\u5404\u5217\u7684\u7ebf\u6027\u7ec4\u5408\u3002\u4e58\u79efAB\u7684\u94f6\u5b50\u7684\u4f4d\u7f6e\u9700\u8981\u8fd9\u6837\u5f3a\u8c03\uff0c\u5373<strong>A\u88abB\u53f3\u4e58\uff0c\u6216B\u88abA\u5de6\u4e58<\/strong>\u3002\u82e5AB=BA\uff0c\u6211\u4eec\u79f0A\u548cB\u5f7c\u6b64<strong>\u53ef\u4ea4\u6362<\/strong>\u3002<\/p>\n<p><span class=\"katex math inline\">{\\color{red}\u8b66\u544a}<\/span><br \/>\n1. \u4e00\u822c\u60c5\u51b5\u4e0b,<span class=\"katex math inline\">AB \\neq BA<\/span><br \/>\n2. \u6d88\u53bb\u5f8b\u5bf9\u77e9\u9635\u4e58\u6cd5\u4e0d\u6210\u7acb\uff0c\u5373\u82e5AB=AC\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\uff0cB=C\u5e76\u4e0d\u6210\u7acb<br \/>\n3. \u82e5\u4e58\u79efAB\u662f\u96f6\u77e9\u9635\uff0c\u4e00\u822c\u60c5\u51b5\u4e0b\uff0c\u4e0d\u80fd\u65ad\u5b9aA=0\u6216B=0<\/p>\n<h2><span id=\"i-4\">\u77e9\u9635\u7684\u4e58\u5e42<\/span><\/h2>\n<p>\u82e5A\u662fnxn\u77e9\u9635\uff0cK\u662f\u6b63\u6574\u6570\uff0c\u5219<span class=\"katex math inline\">A^k<\/span>\u8868\u793ak\u4e2aA\u7684\u4e58\u79ef<\/p>\n<div class=\"katex math multi-line no-emojify\">A^k=\\underbrace{A \u3000\\dots\u3000A}_{k\u4e2a}\n<\/div>\n<p>\u82e5A\u4e0d\u662f\u96f6\u77e9\u9635\uff0c\u4e14x\u5c5e\u4e8e<span class=\"katex math inline\">R^n<\/span>,\u5219<span class=\"katex math inline\">A^kx<\/span>\u8868\u793ax\u88abA\u8fde\u7eed\u5de6\u4e58k\u6b21\u3002\u82e5k=0,\u5219<span class=\"katex math inline\">A^0x<\/span>\u5c31\u662fx\u672c\u8eab\u3002\u56e0\u6b64<span class=\"katex math inline\">A^0<\/span>\u88ab\u89e3\u91ca\u4e3a\u5355\u4f4d\u77e9\u9635\u3002<\/p>\n<h2><span id=\"i-5\">\u77e9\u9635\u7684\u8f6c\u7f6e<\/span><\/h2>\n<p>\u7ed9\u5b9amxn\u77e9\u9635A\uff0c\u5219A\u7684\u8f6c\u7f6e\u662f\u4e00\u4e2anxm\u77e9\u9635\uff0c\u7528<span class=\"katex math inline\">A^T<\/span>\u8868\u793a\uff0c\u5b83\u7684\u5217\u65f6\u7531A\u7684\u5bf9\u5e94\u884c\u6784\u6210\u7684\u3002<\/p>\n<blockquote><p>\n  <strong>\u5b9a\u74063<\/strong> \u8bbeA\u4e0eB\u8868\u793a\u77e9\u9635\uff0c\u5176\u7ef4\u6570\u4f7f\u4e0b\u5217\u548c\u4e0e\u79ef\u6709\u5b9a\u4e49\uff0c\u5219<br \/>\n  a.\u3000<span class=\"katex math inline\">\uff08\uff21^T\uff09^T=A<\/span><br \/>\n  b.\u3000<span class=\"katex math inline\">(A+B)^T=A^T+B^T<\/span><br \/>\n  c.\u3000\u5bf9\u4efb\u610f\u6570\uff52\uff0c<span class=\"katex math inline\">(rA)^T=rA^T<\/span><br \/>\n  d.\u3000<span class=\"katex math inline\">(AB)^T=B^TA^T<\/span><\/p>\n<p>  \u82e5\u5e72\u4e2a\u77e9\u9635\u7684\u4e58\u79ef\u7684\u8f6c\u7f6e\u7b49\u4e8e\u4ed6\u4eec\u7684\u8f6c\u7f6e\u7684\u4e58\u79ef\uff0c\u4f46\u76f8\u4e58\u7684\u987a\u5e8f\u76f8\u53cd\u3002\n<\/p><\/blockquote>\n","protected":false},"excerpt":{"rendered":"<p>Contents1 2.1 \u77e9\u9635\u8fd0\u7b971.1 \u548c\u4e0e\u6807\u91cf\u4e58\u6cd51.2 \u77e9\u9635\u4e58\u6cd51.3 \u77e9\u9635\u4e58\u6cd5\u7684\u6027\u8d281.4 \u77e9\u9635\u7684\u4e58 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":348,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[21,22,26],"tags":[],"class_list":["post-252","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-21","category-22","category-26"],"_links":{"self":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/252","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=252"}],"version-history":[{"count":7,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/252\/revisions"}],"predecessor-version":[{"id":259,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/posts\/252\/revisions\/259"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=\/wp\/v2\/media\/348"}],"wp:attachment":[{"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=252"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=252"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.wayln.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=252"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}