\u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u4e00\u7ec4\u65b9\u7a0b\uff1a<\/p>\n
$$
\n\\begin{eqnarray}
\n-3x_1 + 2x_2 - x_3 = -1 \\newline
\n6x_1 - 6x_2 + 7x_3 = -7 \\newline
\n3x_1 - 4x_2 + 4x_3 = -6
\n\\end{eqnarray}
\n$$<\/p>\n
\u6211\u4eec\u5c06\u5176\u4ee5\u77e9\u9635\u7684\u5f62\u5f0f\u63cf\u8ff0\uff1a<\/p>\n
$$
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{6} & {-6} & {7} \\newline
\n{3} & {-4} & {4}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{x_1} \\newline
\n{x_2} \\newline
\n{x_3}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} \\newline
\n{-7} \\newline
\n{-6}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\u6211\u4eec\u4e5f\u53ef\u4ee5\u5c06\u5176\u8bb0\u4f5c\uff1a$\\rm Ax = b$\u3002<\/p>\n
\u7528\u4e8e\u6c42\u89e3\u7ebf\u6027\u65b9\u7a0b\u7ec4\u7684\u6807\u51c6\u6570\u503c\u7b97\u6cd5\u88ab\u79f0\u4e3a\u9ad8\u65af\u6d88\u5143\u6cd5(Gaussian elimination)<\/strong>\u3002\u6211\u4eec\u9996\u5148\u5c06$\\rm A$\u548c$\\rm b$\u7528\u4e00\u79cd\u53eb\u589e\u5e7f\u77e9\u9635(augmented matrix)<\/strong>\u6765\u8868\u793a\uff1a<\/p>\n $$ \u6211\u4eec\u5c06\u7b2c$i$\u884c\u7684$a_{ii}$\u9009\u4e3a\u4e3b\u5143\u6765\u6d88\u53bb\u5176\u4ed6\u7684\u884c\u4e0a\u7b2c$i$\u5217\u7684\u5143\u7d20\u3002\u6211\u4eec\u9996\u5148\u4ece$a_{11}$\u5f00\u59cb\uff0c\u7528\u5b83\u6d88\u53bb\u7b2c2\u30013\u884c\u7684\u5143\u7d20\uff1a<\/p>\n $$ \u6211\u4eec\u73b0\u5728\u9009\u4e2d$a_{22}$\uff0c\u7528\u5b83\u6d88\u53bb\u7b2c1\u30013\u884c\u7684\u5143\u7d20\uff1a<\/p>\n $$ \u6211\u4eec\u73b0\u5728\u9009\u4e2d$a_{33}$\uff0c\u7528\u5b83\u6d88\u53bb\u7b2c1\u30012\u884c\u7684\u5143\u7d20\uff1a<\/p>\n $$ \u5316\u7b80\u4e00\u4e0b\u5f97\u5230\uff1a<\/p>\n $$ \u5373\uff1a<\/p>\n $$ \u9ad8\u65af\u6d88\u5143\u6cd5\u8fd8\u53ef\u4ee5\u7528\u4e8e\u77e9\u9635\u7684\u6c42\u9006\uff1a<\/p>\n $$ \u5047\u8bbe\u73b0\u5728\u6211\u4eec\u6709\u5982\u4e0b\u77e9\u9635\uff1a<\/p>\n $$ \u5176\u6c42\u9006\u9635\u8fc7\u7a0b\u5982\u4e0b\uff1a<\/p>\n $$ \u4e8e\u662f\u6211\u4eec\u5f97\u5230\u4e86$\\rm A^{-1}$\uff1a<\/p>\n $$ \u5728\u4e4b\u524d\u7684\u4f8b\u5b50\u4e2d\uff1a<\/p>\n $$ \u8fd9\u76f8\u5f53\u4e8e\u628a\u7b2c\u4e00\u884c\u7684\u4e8c\u500d\u52a0\u5230\u7b2c\u4e8c\u884c\uff0c\u56e0\u6b64\u6211\u4eec\u53ef\u4ee5\u7528\u4e00\u4e2a\u77e9\u9635\u6765\u8868\u793a$\\rm M_1$\uff1a<\/p>\n $$ $\\rm M_1$\u57fa\u4e8e\u5355\u4f4d\u77e9\u9635\uff0c\u4e3a\u4e86\u8868\u793a\u201c\u628a\u7b2c\u4e00\u884c\u76842\u500d\u52a0\u5230\u7b2c\u4e8c\u884c\u201d\uff0c\u56e0\u6b64$m_{2,1}$=2\u3002\u540c\u7406\uff0c\u6211\u4eec\u7ee7\u7eed\u6765\u770b\u8fd9\u4e00\u4e2a\u53d8\u6362\uff1a<\/p>\n $$ \u8fd9\u76f8\u5f53\u4e8e\u201c\u5c06\u7b2c\u4e00\u884c\u76841\u500d\u52a0\u5230\u7b2c\u4e09\u884c\u201d\uff0c\u56e0\u6b64$m_{3,1}=1$\uff1a<\/p>\n $$ \u540c\u7406\uff0c\u6211\u4eec\u518d\u6765\u770b\u4e00\u4e2a\u53d8\u6362\uff1a<\/p>\n $$ \u8fd9\u76f8\u5f53\u4e8e\u201c\u5c06\u7b2c\u4e8c\u884c\u7684-1\u500d\u52a0\u5230\u7b2c\u4e09\u884c\u201d\uff0c\u56e0\u6b64$m_{3,2}=-1$\u3002<\/p>\n $$ \u4e8e\u662f\u6211\u4eec\u53d1\u73b0\uff1a<\/p>\n $$ \u5176\u4e2d$\\rm U$\u662f\u4e0a\u4e09\u89d2\u77e9\u9635\uff0c\u6211\u4eec\u5c06\u5728\u4e0b\u4e00\u8282\u5177\u4f53\u8ba8\u8bba\u3002<\/p>\n \u5728\u7b2c\u4e09\u8282\u4e2d\uff0c\u6211\u4eec\u5f97\u5230\u4e86\uff1a<\/p>\n $$ \u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u5199\u4f5c\uff1a<\/p>\n $$ \u540c\u65f6\u6211\u4eec\u53d1\u73b0\uff1a<\/p>\n $$ $$ $$ $$ \u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u4f5c\uff1a<\/p>\n $$ \u56de\u987e\u6211\u4eec\u4e4b\u524d\u8ba1\u7b97\u65b9\u7a0b\uff1a<\/p>\n $$ \u4e3a\u4e86\u63d0\u9ad8\u8ba1\u7b97\u673a\u7684\u8ba1\u7b97\u901f\u5ea6\uff0c\u6211\u4eec\u5f15\u5165$\\rm LU$\u6765\u8868\u793a$A$\uff0c\u56e0\u6b64\u65b9\u7a0b\u53d8\u6210\u4e86\uff1a<\/p>\n $$ \u6211\u4eec\u5c06$\\rm Ux$\u8bb0\u4f5c$\\rm y$\uff0c\u4e8e\u662f\u89e3\u65b9\u7a0b\u7684\u6b65\u9aa4\u53d8\u6210\u4e86\uff1a<\/p>\n $$ \u6211\u4eec\u4f9d\u7136\u901a\u8fc7\u4e4b\u524d\u7684\u4f8b\u5b50\u6765\u770b\u4e00\u4e0b\u8fd9\u4e2a\u89e3\u65b9\u7a0b\u7684\u8fc7\u7a0b\uff1a<\/p>\n $$ \u9996\u5148\u6211\u4eec\u4ee4$\\rm y = Ux$\uff0c\u7136\u540e\u89e3\u65b9\u7a0b$\\rm Ly = b$\uff1a<\/p>\n $$ \u63a5\u7740\uff0c\u6211\u4eec\u518d\u89e3\u65b9\u7a0b$\\rm Ux = y$\uff1a<\/p>\n $$ \u4e00\u3001\u9ad8\u65af\u6d88\u5143\u6cd5 \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u4e00\u7ec4\u65b9\u7a0b\uff1a $$ \\begin{eqnarray} -3x_1 + 2x_2 …<\/p>","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[293],"tags":[],"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1362"}],"collection":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/comments?post=1362"}],"version-history":[{"count":6,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1362\/revisions"}],"predecessor-version":[{"id":1368,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1362\/revisions\/1368"}],"wp:attachment":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/media?parent=1362"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/categories?post=1362"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/tags?post=1362"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} & {-1} \\newline
\n{6} & {-6} & {7} & {-7} \\newline
\n{3} & {-4} & {4} & {-6}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} & {-1} \\newline
\n{0} & {-2} & {5} & {-9} \\newline
\n{0} & {-2} & {3} & {-7}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {0} & {4} & {-10} \\newline
\n{0} & {-2} & {5} & {-9} \\newline
\n{0} & {0} & {-2} & {2}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {0} & {0} & {-6} \\newline
\n{0} & {-2} & {0} & {-4} \\newline
\n{0} & {0} & {1} & {-1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} & {2} \\newline
\n{0} & {1} & {0} & {2} \\newline
\n{0} & {0} & {1} & {-1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\begin{eqnarray}
\nx_1 &=& 2 \\newline
\nx_2 &=& 2 \\newline
\nx_3 &=& -1
\n\\end{eqnarray}
\n$$<\/p>\n\u4e8c\u3001\u6c42\u9006<\/h2>\n
\n\\rm AI \\to I{A}^{-1}
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{6} & {-6} & {7} \\newline
\n{3} & {-4} & {4}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\begin{eqnarray}
\n& \\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} & {1} & {0} & {0} \\newline
\n{6} & {-6} & {7} & {0} & {1} & {0} \\newline
\n{3} & {-4} & {4} & {0} & {0} & {1}
\n\\end{array}
\n\\right) \\newline
\n& \\Downarrow \\newline
\n& \\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} & {1} & {0} & {0} \\newline
\n{0} & {-2} & {5} & {2} & {1} & {0} \\newline
\n{0} & {-2} & {3} & {1} & {0} & {1}
\n\\end{array}
\n\\right) \\newline
\n& \\Downarrow \\newline
\n& \\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {0} & {4} & {3} & {1} & {0} \\newline
\n{0} & {-2} & {5} & {2} & {1} & {0} \\newline
\n{0} & {0} & {-2} & {-1} & {-1} & {1}
\n\\end{array}
\n\\right) \\newline
\n& \\Downarrow \\newline
\n& \\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {0} & {0} & {1} & {-1} & {2} \\newline
\n{0} & {-2} & {0} & {-1\/2} & {-3\/2} & {5\/2} \\newline
\n{0} & {0} & {-2} & {-1} & {-1} & {1}
\n\\end{array}
\n\\right) \\newline
\n& \\Downarrow \\newline
\n& \\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} & {-1\/3} & {1\/3} & {-2\/3} \\newline
\n{0} & {1} & {0} & {1\/4} & {3\/4} & {-5\/4} \\newline
\n{0} & {0} & {1} & {1\/2} & {1\/2} & {-1\/2}
\n\\end{array}
\n\\right) \\newline
\n\\end{eqnarray}
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-1\/3} & {1\/3} & {-2\/3} \\newline
\n{1\/4} & {3\/4} & {-5\/4} \\newline
\n{1\/2} & {1\/2} & {-1\/2}
\n\\end{array}
\n\\right)
\n$$<\/p>\n\u4e09\u3001\u521d\u7b49\u53d8\u6362<\/h2>\n
\n{\\rm A} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{6} & {-6} & {7} \\newline
\n{3} & {4} & {4}
\n\\end{array}
\n\\right) \\to
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{3} & {4} & {4}
\n\\end{array}
\n\\right) =
\n{\\rm M_{1}A}
\n$$<\/p>\n
\n{\\rm M_1} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{2} & {1} & {0} \\newline
\n{0} & {0} & {1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{3} & {4} & {4}
\n\\end{array}
\n\\right) \\to
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{0} & {-2} & {3}
\n\\end{array}
\n\\right) =
\n{\\rm {M}_{2}{M}_{1}{A}}
\n$$<\/p>\n
\n{\\rm M_2} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{0} & {1} & {0} \\newline
\n{1} & {0} & {1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{0} & {-2} & {3}
\n\\end{array}
\n\\right) \\to
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{0} & {0} & {-2}
\n\\end{array}
\n\\right) =
\n{\\rm {M}_{3}{M}_{2}{M}_{1}{A}}
\n$$<\/p>\n
\n{\\rm M_3} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{0} & {1} & {0} \\newline
\n{0} & {-1} & {1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\rm {M}_{3}{M}_{2}{M}_{1}A = U =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{0} & {0} & {-2}
\n\\end{array}
\n\\right)
\n$$<\/p>\n\u56db\u3001LU\u5206\u89e3<\/h2>\n
\n\\rm {M}_{3}{M}_{2}{M}_{1}A = U
\n$$<\/p>\n
\n\\rm A = {M}_{1}^{-1}{M}_{2}^{-1}{M}_{3}^{-1}U
\n$$<\/p>\n
\n{\\rm M_1} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{2} & {1} & {0} \\newline
\n{0} & {0} & {1}
\n\\end{array}
\n\\right),
\n{\\rm {M}_{1}^{-1}} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{-2} & {1} & {0} \\newline
\n{0} & {0} & {1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm M_2} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{0} & {1} & {0} \\newline
\n{1} & {0} & {1}
\n\\end{array}
\n\\right),
\n{\\rm {M}_{2}^{-1}} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{0} & {1} & {0} \\newline
\n{-1} & {0} & {1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm M_2} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{0} & {1} & {0} \\newline
\n{0} & {-1} & {1}
\n\\end{array}
\n\\right),
\n{\\rm {M}_{2}^{-1}} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{0} & {1} & {0} \\newline
\n{0} & {1} & {1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n{\\rm {M}_{1}^{-1}{M}_{2}^{-1}{M}_{3}^{-1}} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{-2} & {1} & {0} \\newline
\n{-1} & {1} & {1}
\n\\end{array}
\n\\right)
\n= {\\rm L}
\n$$<\/p>\n
\n\\rm LU = A
\n$$<\/p>\n\u4e94\u3001(LU)x = b<\/h2>\n
\n\\rm Ax = b
\n$$<\/p>\n
\n\\rm LUx = b
\n$$<\/p>\n
\n\\begin{eqnarray}
\n&1&: \\quad \\rm let \\ y = Ux \\newline
\n&2&: \\quad \\rm solve \\ Ly = b \\newline
\n&3&: \\quad \\rm slove \\ Ux = y
\n\\end{eqnarray}
\n$$<\/p>\n
\n{\\rm L} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{-2} & {1} & {0} \\newline
\n{-1} & {1} & {1}
\n\\end{array}
\n\\right),
\n{\\rm U} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{0} & {0} & {-2}
\n\\end{array}
\n\\right),
\n{\\rm b} =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-1} \\newline
\n{-7}\\newline
\n{-6}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{1} & {0} & {0} \\newline
\n{-2} & {1} & {0} \\newline
\n{-1} & {1} & {1}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{y_1} \\newline
\n{y_2} \\newline
\n{y_3}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-1} \\newline
\n{-7}\\newline
\n{-6}
\n\\end{array}
\n\\right)
\n\\to
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{y_1} \\newline
\n{y_2} \\newline
\n{y_3}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-1} \\newline
\n{-9}\\newline
\n{-2}
\n\\end{array}
\n\\right)
\n$$<\/p>\n
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-3} & {2} & {-1} \\newline
\n{0} & {-2} & {5} \\newline
\n{0} & {0} & {-2}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{x_1} \\newline
\n{x_2} \\newline
\n{x_3}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{-1} \\newline
\n{-9}\\newline
\n{-2}
\n\\end{array}
\n\\right)
\n\\to
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{x_1} \\newline
\n{x_2} \\newline
\n{x_3}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{2} \\newline
\n{2}\\newline
\n{-1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n","protected":false},"excerpt":{"rendered":"