{"id":2137,"date":"2023-04-02T10:06:28","date_gmt":"2023-04-02T02:06:28","guid":{"rendered":"https:\/\/www.appblog.cn\/?p=2137"},"modified":"2023-04-05T19:59:59","modified_gmt":"2023-04-05T11:59:59","slug":"fundamentals-of-advanced-mathematics-linear-algebra-and-matrix","status":"publish","type":"post","link":"https:\/\/www.appblog.cn\/index.php\/2023\/04\/02\/fundamentals-of-advanced-mathematics-linear-algebra-and-matrix\/","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66\u57fa\u7840\uff1a\u7ebf\u6027\u4ee3\u6570\u4e0e\u77e9\u9635"},"content":{"rendered":"

\u7ebf\u6027\u4ee3\u6570\u4e0e\u77e9\u9635<\/h2>\n

\u7ebf\u6027\u4ee3\u6570<\/strong>\uff1a\u7ebf\u6027\u4ee3\u6570\u4e2d\u7684\u57fa\u672c\u91cf\u6307\u7684\u662f\u5411\u91cf\uff0c\u57fa\u672c\u5173\u7cfb\u662f\u4e25\u683c\u7684\u7ebf\u6027\u5173\u7cfb\u3002\u4e5f\u5c31\u662f\u53ef\u4ee5\u7b80\u5355\u7684\u5c06\u7ebf\u6027\u4ee3\u6570\u7406\u89e3\u4e3a\u5411\u91cf\u4e0e\u5411\u91cf\u4e4b\u95f4\u7684\u7ebf\u6027\u5173\u7cfb\u7684\u6620\u5c04\u3002<\/p>\n

\u77e9\u9635<\/strong>\uff1a\u63cf\u8ff0\u7ebf\u6027\u4ee3\u6570\u4e2d\u7ebf\u6027\u5173\u7cfb\u7684\u53c2\u6570\uff0c\u5373\u77e9\u9635\u662f\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u53ef\u4ee5\u5c06\u4e00\u4e9b\u5411\u91cf\u8f6c\u6362\u4e3a\u53e6\u4e00\u4e9b\u5411\u91cf\u3002<\/p>\n

<\/p>\n

\u77e9\u9635\u8868\u793a\uff1a\u6570\u57df$F$\u4e2d$m \\ast n$\u4e2a\u6570\u6392\u6210$m$\u884c$n$\u5217\uff0c\u5e76\u62ec\u4ee5\u5706\u62ec\u5f27\uff08\u6216\u65b9\u62ec\u5f27\uff09\u7684\u6570\u8868\u793a\u79f0\u4e3a\u6570\u57df$F$\u4e0a\u7684\u77e9\u9635\uff0c\u901a\u5e38\u7528\u5927\u5199\u5b57\u6bcd\u8bb0\u4f5c$A$\u6216\u8005$A{m \\ast n}$\uff0c\u6709\u65f6\u4e5f\u8bb0\u4f5c$A=(a<\/em>{ij}){m \\ast n}(i=1,2,…m;n=1,2,…n)$\uff0c\u5176\u4e2d$a<\/em>{ij}$\u8868\u793a\u77e9\u9635$A$\u7684\u7b2c$i$\u884c\u548c\u7b2c$j$\u5217\u5143\u7d20\u3002\u5f53$F$\u4e3a\u5b9e\u6570\u57df$R$\u65f6\uff0c$A$\u53eb\u505a\u5b9e\u77e9\u9635\uff1b\u5f53$F$\u4e3a\u590d\u6570\u57df$C$\u65f6\uff0c$A$\u53eb\u505a\u590d\u77e9\u9635\u3002<\/p>\n

$$
\n\\begin {pmatrix}
\na{11} & a<\/em>{12} & \\cdots & a{1n} \\
\na<\/em>{21} & a{22} & \\cdots & a<\/em>{2n} \\
\n\\vdots & \\vdots & \\ddots & \\vdots \\
\na{m1} & a<\/em>{m2} & \\cdots & a_{mn}
\n\\end {pmatrix}
\n$$<\/p>\n

\u5f53$m=1$\u6216$n=1$\u65f6\uff0c\u79f0$A$\u4e3a\u884c\u5411\u91cf<\/strong>\u6216\u5217\u5411\u91cf<\/strong><\/p>\n

$$
\nA = \\begin {pmatrix} a{1} & a<\/em>{1} & \\cdots & a{n} \\end {pmatrix} \\,\\,\u6216\\,\\,
\nA =
\n\\begin {pmatrix}
\na<\/em>{1} \\
\na{2} \\
\n\\vdots \\
\na<\/em>{m}
\n\\end {pmatrix}
\n$$<\/p>\n

\u77e9\u9635\u76f8\u7b49<\/strong>\uff1a\u5bf9\u4e8e\u4e24\u4e2a\u77e9\u9635$A$\u548c$B$\uff0c\u5f53\u5b83\u4eec\u7684\u884c\u6570\u76f8\u540c\uff0c\u5217\u6570\u76f8\u540c\uff0c\u5e76\u4e14\u5bf9\u5e94\u4f4d\u7f6e\u4e0a\u7684\u5143\u7d20\u90fd\u76f8\u7b49\u65f6\uff0c\u79f0\u77e9\u9635$A$\u4e0e$B$\u76f8\u7b49\uff0c\u8bb0\u4f5c$A=B$<\/p>\n

\u540c\u578b\u77e9\u9635<\/strong>\uff1a\u82e5\u4e24\u4e2a\u77e9\u9635\u884c\u6570\u4e0e\u5217\u6570\u5206\u522b\u76f8\u7b49\uff0c\u5219\u4e3a\u540c\u578b\u77e9\u9635<\/p>\n

\u65b9\u9635<\/strong>\uff1a\u5982\u679c\u77e9\u9635$A$\u4e2d$m$\u7b49\u4e8e$n$\uff0c\u90a3\u4e48\u79f0\u77e9\u9635$A$\u4e3a$n$\u9636\u77e9\u9635\uff08\u6216$n$\u9636\u65b9\u9635\uff09\u3002\u4ece\u5de6\u4e0a\u5230\u53f3\u4e0b\u7684\u5bf9\u89d2\u7ebf\u79f0\u4e3a\u4e3b\u5bf9\u89d2\u7ebf\uff0c\u4ece\u53f3\u4e0a\u5230\u5de6\u4e0b\u7684\u5bf9\u89d2\u7ebf\u79f0\u4e3a\u6b21\u5bf9\u89d2\u7ebf<\/p>\n

\u8d1f\u77e9\u9635<\/strong>\uff1a\u5bf9\u4e8e\u77e9\u9635$A$\uff08\u6240\u6709\u5143\u7d20$m \\ast n$\uff09\uff0c\u5404\u4e2a\u5143\u7d20\u53d6\u76f8\u53cd\u6570\u5f97\u5230\u7684\u77e9\u9635\u79f0\u4e3a$A$\u7684\u8d1f\u77e9\u9635\uff0c\u8bb0\u4e3a$-A$<\/p>\n

\u4e0a\u4e09\u89d2\u9635<\/strong>\uff1a\u4e3b\u5bf9\u89d2\u7ebf\uff08\u4e0d\u542b\uff09\u4ee5\u4e0b\u7684\u5143\u7d20\u5747\u4e3a0\u7684\u77e9\u9635\u79f0\u4e3a\u4e0a\u4e09\u89d2\u9635\uff0c\u4e3b\u5bf9\u89d2\u7ebf\uff08\u4e0d\u542b\uff09\u4ee5\u4e0a\u7684\u5143\u7d20\u5747\u4e3a0\u7684\u77e9\u9635\u79f0\u4e3a\u4e0b\u4e09\u89d2\u9635\uff0c<\/p>\n

\u5bf9\u89d2\u77e9\u9635<\/strong>\uff1a\u65e2\u662f\u4e0a\u4e09\u89d2\u9635\uff0c\u53c8\u662f\u4e0b\u4e09\u89d2\u9635\uff08\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e0d\u5168\u4e3a0\uff0c\u5176\u4f59\u5143\u7d20\u5747\u4e3a0\uff09\u7684\u77e9\u9635\u79f0\u4e3a\u5bf9\u89d2\u65b9\u9635\u6216\u5bf9\u89d2\u77e9\u9635\uff0c\u8bb0\u4e3a$D = \\operatorname{diag}\\begin {pmatrix} a{11} & a<\/em>{22} & \\cdots & a_{nn} \\end {pmatrix}$<\/p>\n

\u77e9\u9635\u7684\u8ff9<\/strong>\uff1a\u5bf9\u89d2\u5143\u7d20\u7684\u548c$\\sum{i=1}^n a<\/em>{ii}$\uff0c\u79f0\u4e3a\u65b9\u9635$A$\u7684\u8ff9\uff0c\u8bb0\u4e3a$\\operatorname{tr}A$<\/p>\n

\u5355\u4f4d\u77e9\u9635<\/strong>\uff1a$n$\u9636\u65b9\u9635\u4e2d\u9664\u4e86\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u5143\u7d20\u5916\uff0c\u5176\u5b83\u5143\u7d20\u5747\u4e3a0\uff0c\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u5747\u4e3a1\uff0c\u90a3\u4e48\u6b64\u65f6\u7684$n$\u9636\u65b9\u9635\u53eb\u505a$n$\u9636\u5355\u4f4d\u77e9\u9635\u3002\u5355\u4f4d\u77e9\u9635\u5e38\u7528$E$\u6216\u8005$I$\u8868\u793a<\/p>\n

$$
\nE = \\begin {pmatrix}
\n1 & 0 & \\cdots & 0 \\
\n0 & 1 & \\cdots & 0 \\
\n\\vdots & \\vdots & \\ddots & \\vdots \\
\n0 & 0 & \\cdots & 1
\n\\end {pmatrix}
\n$$<\/p>\n

\u96f6\u77e9\u9635<\/strong>\uff1a\u5982\u679c\u77e9\u9635$A$\u4e2d\u7684\u6240\u6709\u5143\u7d20$m \\ast n$\u5747\u4e3a0\uff0c\u90a3\u4e48\u6b64\u65f6\u77e9\u9635$A$\u53eb\u505a\u96f6\u77e9\u9635\uff0c\u53ef\u4ee5\u8bb0\u4e3a$0$<\/p>\n

\u77e9\u9635\u8fd0\u7b97<\/h2>\n","protected":false},"excerpt":{"rendered":"

\u7ebf\u6027\u4ee3\u6570\u4e0e\u77e9\u9635 \u7ebf\u6027\u4ee3\u6570\uff1a\u7ebf\u6027\u4ee3\u6570\u4e2d\u7684\u57fa\u672c\u91cf\u6307\u7684\u662f\u5411\u91cf\uff0c\u57fa\u672c\u5173\u7cfb\u662f\u4e25\u683c\u7684\u7ebf\u6027\u5173\u7cfb\u3002\u4e5f\u5c31\u662f\u53ef\u4ee5\u7b80\u5355\u7684\u5c06\u7ebf\u6027\u4ee3\u6570\u7406 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[509],"tags":[],"class_list":["post-2137","post","type-post","status-publish","format-standard","hentry","category-mathematics-fundamentals"],"_links":{"self":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2137","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/comments?post=2137"}],"version-history":[{"count":0,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/posts\/2137\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/media?parent=2137"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/categories?post=2137"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.appblog.cn\/index.php\/wp-json\/wp\/v2\/tags?post=2137"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}