6. \u9006\u77e9\u9635\u3001\u5217\u7a7a\u95f4\u548c\u96f6\u7a7a\u95f4
<\/p>\n\n\n\n
\u7cfb\u6570\u77e9\u9635 $ \\boldsymbol{A} $ \uff0c\u672a\u77e5\u6570\u5411\u91cf $\\bf\\vec{x} $ \uff0c\u53f3\u4fa7\u5e38\u6570\u5411\u91cf $ \\bf\\vec{v} $.<\/p>\n\n\n\n
$ \\boldsymbol{A} \\bf\\vec{x} = \\bf\\vec{v}$<\/p>\n\n\n\n
\u5176\u4e2d\u7cfb\u6570\u77e9\u9635 $ \\boldsymbol{A} $ \u4ee3\u8868\u4e86\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u5bf9\u8be5\u65b9\u7a0b\u7684\u6c42\u89e3\u8868\u793a\u9700\u8981\u627e\u5230\u4e00\u4e2a\u5411\u91cf $\\bf\\vec{x} $ \uff0c\u8ba9\u5b83\u5728\u7ecf\u8fc7\u53d8\u6362\u540e\u5f97\u5230\u5411\u91cf $\\bf\\vec{v} $.<\/p>\n\n\n\n
\u5982\u679c $ \\boldsymbol{A} $ \u6ca1\u6709\u628a\u7a7a\u95f4\u538b\u7f29\u81f3\u4f4e\u7ef4\uff08\u4e5f\u5c31\u662f\u884c\u5217\u5f0f\u4e0d\u4e3a $0$ \uff0c\u6211\u4eec\u53ef\u4ee5\u627e\u5230\u4e00\u4e2a\u5411\u91cf $\\bf\\vec{x} $ \uff0c\u8ba9\u5b83\u5728\u7ecf\u8fc7\u53d8\u6362\u540e\u5f97\u5230\u5411\u91cf $\\bf\\vec{v} $ . \u6211\u4eec\u53ea\u8981\u5bf9 $\\bf\\vec{v} $ \u5c06\u8fd9\u79cd\u7ebf\u6027\u53d8\u6362\u201c\u5012\u5e26\u201d\u56de\u53bb\uff0c\u5c31\u80fd\u5f97\u5230\u65b9\u7a0b\u7ec4\u7684\u89e3 $\\bf\\vec{x} $ \u4e86\u3002 \u8fd9\u79cd\u9006\u5411\u7684 \u201c\u5012\u5e26\u201d \u5bf9\u5e94\u4e86\u53e6\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\uff0c\u8bb0\u4f5c $ \\boldsymbol{A}^{-1} $\uff0c\u5373 $ \\boldsymbol{A} $ \u7684\u9006\u3002<\/p>\n\n\n\n
\u6bd4\u5982\uff0c\u5047\u8bbe $ \\boldsymbol{A} $ \u662f\u5c06\u5e73\u9762\u987a\u65f6\u9488\u65cb\u8f6c $90\u00b0$\uff0c\u90a3\u4e48 $ \\boldsymbol{A}^{-1} $ \u5219\u4ee3\u8868\u5c06\u5e73\u9762\u9006\u65f6\u9488\u65cb\u8f6c $90\u00b0$.<\/p>\n\n\n\n
\u8fd9\u91cc\uff0c $\\boldsymbol{A} =
\\begin{bmatrix}
0 & -1 \\newline
1 & 0
\\end{bmatrix} $$ \uff0c$$\\boldsymbol{A}^{-1} =
\\begin{bmatrix}
0 & 1 \\newline
-1 & 0
\\end{bmatrix} $ .<\/p>\n\n\n\n
\u53c8\u6bd4\u5982\uff0c\u5047\u8bbe $ \\boldsymbol{A} $ \u662f\u5c06 $\\hat{j}$ \u5411\u53f3\u526a\u5207\u79fb\u52a8 $1$ \u4e2a\u5355\u4f4d\uff0c\u90a3\u4e48 $ \\boldsymbol{A}^{-1} $ \u5219\u4ee3\u8868\u5c06 $\\hat{j}$ \u5411\u5de6\u526a\u5207\u79fb\u52a8 $1$ \u4e2a\u5355\u4f4d\u3002<\/p>\n\n\n\n
\u8fd9\u91cc\uff0c $\\boldsymbol{A} =
\\begin{bmatrix}
1 & 1 \\newline
0 & 1
\\end{bmatrix} $ \uff0c$\\boldsymbol{A}^{-1} =
\\begin{bmatrix}
1 & -1 \\newline
0 & 1
\\end{bmatrix} $ .<\/p>\n\n\n\n
$ \\boldsymbol{A}^{-1}\\boldsymbol{A}$ \u8868\u793a\u4ec0\u4e48\u4e5f\u6ca1\u505a\uff0c\u8fd9\u79cd\u53d8\u6362\u88ab\u79f0\u4e3a\u6052\u7b49\u53d8\u6362\u3002<\/p>\n\n\n\n
\u6240\u4ee5\u6c42\u89e3$ \\boldsymbol{A} \\bf\\vec{x} = \\bf\\vec{v}$\uff0c\u53ef\u4ee5\u5728\u65b9\u7a0b\u5de6\u53f3\u4e24\u8fb9\u540c\u65f6\u5de6\u4e58 $ \\boldsymbol{A}^{-1} $ \uff0c\u5f97\u5230 $\\bf\\vec{x} = \\boldsymbol{A}^{-1} \\bf\\vec{v}$ .<\/p>\n\n\n\n
\u5728\u51e0\u4f55\u4e0a\uff0c\u53ef\u4ee5\u7406\u89e3\u4e3a\u5bf9 $\\bf\\vec{v} $ \u505a\u4e86\u9006\u5411\u7684\u53d8\u6362 $ \\boldsymbol{A}^{-1} $ .<\/p>\n\n\n\n
\u5982\u679c $ \\det(\\boldsymbol{A})$ \u4e3a\u96f6\uff0c\u4e5f\u5c31\u662f\u77e9\u9635 $ \\boldsymbol{A} $ \u5c06\u7a7a\u95f4\u538b\u7f29\u5230\u4e86\u4f4e\u7ef4\u5ea6\uff0c\u90a3\u4e48\u5c31\u4e0d\u5b58\u5728 $ \\boldsymbol{A}^{-1} $\uff0c\u4e0d\u80fd\u628a\u4e00\u6761\u7ebf\u89e3\u538b\u6210\u4e00\u4e2a\u5e73\u9762\u3002<\/p>\n\n\n\n
\u4f46\u662f\uff0c\u5373\u4f7f $ \\det(\\boldsymbol{A})$ \u4e3a\u96f6\uff0c\u65b9\u7a0b\u7ec4\u4ecd\u7136\u6709\u53ef\u80fd\u6709\u89e3\u3002\u6bd4\u5982\uff0c\u77e9\u9635 $ \\boldsymbol{A} $ \u5c06\u4e09\u7ef4\u7a7a\u95f4\u538b\u6210\u4e86\u4e8c\u7ef4\uff0c\u800c $\\bf\\vec{v}$ \u6070\u597d\u843d\u5728\u8fd9\u4e2a\u53d8\u6362\u540e\u7684\u4e8c\u7ef4\u5e73\u9762\u4e0a\u3002<\/p>\n\n\n\n
\u77e9\u9635\u7684\u79e9\uff08rank\uff09\u662f\u7a7a\u95f4\u53d8\u6362\u540e\u7684\u7ef4\u6570\u3002<\/p>\n\n\n\n
\u5f53\u4e00\u4e2a\u77e9\u9635\u628a\u7a7a\u95f4\u538b\u7f29\u4e3a\u4e00\u6761\u7ebf\uff08\u4e00\u7ef4\uff09\u65f6\uff0c\u77e9\u9635\u7684\u79e9\u4e3a $1$.<\/p>\n\n\n\n
\u77e9\u9635\u7684\u5217\u7a7a\u95f4\uff08column space\uff09\u662f\u5176\u5217\u5f20\u6210\u7684\u7a7a\u95f4\uff0c\u4e8e\u662f\uff0c\u66f4\u52a0\u7cbe\u786e\u5b9a\u4e49\u77e9\u9635\u7684\u79e9\u4e3a\uff0c\u5176\u5217\u7a7a\u95f4\u7684\u7ef4\u6570\u3002\u5f53\u77e9\u9635\u7684\u79e9\u548c\u5217\u6570\u76f8\u7b49\u65f6\uff08\u6700\u5927\u60c5\u51b5\uff09\uff0c\u79f0\u77e9\u9635\u662f\u6ee1\u79e9\u7684\uff08full rank\uff09\u3002<\/p>\n\n\n\n
\u5982\u679c\u4e00\u4e2a\u4e8c\u7ef4\u77e9\u9635\u662f\u6ee1\u79e9\u7684\uff0c\u90a3\u4e48\u5728\u7ecf\u8fc7\u7ebf\u6027\u53d8\u6362\u4e4b\u540e\uff0c\u53ea\u6709\u96f6\u5411\u91cf\u548c\u539f\u5148\u7684\u4f4d\u7f6e\u91cd\u5408\u3002\u5982\u679c\u4e00\u4e2a\u4e8c\u7ef4\u77e9\u9635\u5c06\u5e73\u9762\u538b\u6210\u4e86\u7ebf\uff0c\u90a3\u4e48\u5219\u6709\u4e00\u6761\u7ebf\u4e0a\u7684\u70b9\u90fd\u88ab\u538b\u5728\u4e86\u539f\u70b9\u4e0a\u3002<\/p>\n\n\n\n
\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u4e5f\u662f\uff0c\u5982\u679c\u4e09\u7ef4\u77e9\u9635\u7684\u79e9\u4e3a $2$\uff0c\u5219\u8be5\u77e9\u9635\u4f1a\u628a\u7a7a\u95f4\u538b\u6210\u5e73\u9762\uff0c\u5219\u6709\u4e00\u6761\u7ebf\u4e0a\u7684\u70b9\u88ab\u538b\u5728\u539f\u70b9\u4e0a\u3002\u5982\u679c\u4e09\u7ef4\u77e9\u9635\u7684\u79e9\u4e3a $1$\uff0c\u5219\u8be5\u77e9\u9635\u4f1a\u628a\u7a7a\u95f4\u538b\u6210\u76f4\u7ebf\uff0c\u5219\u6709\u4e00\u6574\u4e2a\u5e73\u9762\u4e0a\u7684\u70b9\u88ab\u538b\u5728\u539f\u70b9\u4e0a\u3002<\/p>\n\n\n\n
\u8fd9\u4e9b\u53d8\u6362\u540e\u843d\u5728\u539f\u70b9\u4e0a\u7684\u5411\u91cf\u7684\u96c6\u5408\u88ab\u79f0\u4e3a\u77e9\u9635\u7684\u96f6\u7a7a\u95f4\uff08null space\uff09\u6216\u6838\uff08kernel\uff09\u3002<\/p>\n\n\n\n
\u5bf9\u4e8e\u7ebf\u6027\u65b9\u7a0b\u7ec4 $\\boldsymbol{A} \\bf\\vec{x} = \\bf\\vec{v}$ \u6765\u8bf4\uff0c\u5f53\u5e38\u6570\u9879\u5168\u90e8\u4e3a $0$ \u65f6\uff0c\u77e9\u9635\u7684\u96f6\u7a7a\u95f4\u7ed9\u51fa\u7684\u5c31\u662f\u8fd9\u4e2a\u65b9\u7a0b\u7ec4\u6240\u6709\u53ef\u80fd\u7684\u89e3\uff08\u77e9\u9635 $ \\boldsymbol{A}$ \u628a $\\bf\\vec{x}$ \u538b\u5728\u4e86\u539f\u70b9\u4e0a\uff09\u3002<\/p>\n\n\n\n \n","protected":false},"excerpt":{"rendered":"","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[7,13],"tags":[34,33],"class_list":["post-288","post","type-post","status-publish","format-standard","hentry","category-7","category-13","tag-34","tag-33"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=\/wp\/v2\/posts\/288","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=288"}],"version-history":[{"count":4,"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=\/wp\/v2\/posts\/288\/revisions"}],"predecessor-version":[{"id":292,"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=\/wp\/v2\/posts\/288\/revisions\/292"}],"wp:attachment":[{"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=288"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=288"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.vincedeck.top\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}