{"id":1375,"date":"2024-06-06T15:11:59","date_gmt":"2024-06-06T07:11:59","guid":{"rendered":"https:\/\/swordofmorning.com\/?p=1375"},"modified":"2025-10-09T13:55:01","modified_gmt":"2025-10-09T05:55:01","slug":"linear-algebra-04","status":"publish","type":"post","link":"https:\/\/swordofmorning.com\/index.php\/2024\/06\/06\/linear-algebra-04\/","title":{"rendered":"\u7ebf\u6027\u4ee3\u6570\uff1a\u7279\u5f81\u503c\u4e0e\u7279\u5f81\u5411\u91cf"},"content":{"rendered":"

\u4e00\u3001\u884c\u5217\u5f0f<\/h2>\n

  \u884c\u5217\u5f0f\u7684\u57fa\u672c\u5b9a\u4e49\u5df2\u7ecf\u5728\u300aLinear Algebra 01 Matrix\u300b\u4e2d\u7684\u7b2c\u516d\u7ae0\u201c\u9006\u77e9\u9635\u201d\u4e2d\u7b80\u5355\u8ba8\u8bba\u3002\u8fd9\u91cc\u4e0d\u518d\u8d58\u8ff0\u3002<\/p>\n

1.1 \u57fa\u672c\u6027\u8d28<\/h3>\n

  \u4e0b\u9762\u6211\u4eec\u7b80\u8981\u8bf4\u660e\u4e00\u4e0b\u884c\u5217\u5f0f\u5177\u6709\u7684\u6027\u8d28\uff1a<\/p>\n

    \n
  1. \u884c\u5217\u5f0f\u4e2d\u4e00\u884c(\u5217)\u5168\u4e3a\u96f6\uff0c\u5219\u6b64\u884c\u5217\u5f0f\u7684\u503c\u4e3a0\uff1b<\/li>\n
  2. \u5728\u67d0\u4e00\u884c(\u5217)\u6709\u516c\u56e0\u5b50$k$\uff0c\u5219\u53ef\u4ee5\u5c06\u5176\u63d0\u524d\uff1a$\\left(\\begin{array}{*{20}{c}} {a} & {b} \\newline {kc} & {kd} \\newline \\end{array} \\right) = k \\left(\\begin{array}{*{20}{c}} {a} & {b} \\newline {c} & {d} \\newline \\end{array} \\right)$\uff1b<\/li>\n
  3. \u67d0\u4e00\u884c(\u5217)\u7684\u6bcf\u4e2a\u5143\u7d20\u662f\u4e24\u6570\u4e4b\u548c\uff0c\u53ef\u4ee5\u62c6\u5206\u4e3a\u4e24\u4e2a\u76f8\u52a0\u7684\u884c\u5217\u5f0f\uff1a$\\left(\\begin{array}{*{20}{c}} {a} & {b} \\newline {c + e} & {d + f} \\newline \\end{array} \\right) = \\left(\\begin{array}{*{20}{c}} {a} & {b} \\newline {c} & {d} \\newline \\end{array} \\right) + \\left(\\begin{array}{*{20}{c}} {a} & {b} \\newline {e} & {f} \\newline \\end{array} \\right)$\uff1b<\/li>\n
  4. \u4e24\u884c(\u5217)\u4e92\u6362\uff0c\u6539\u53d8\u884c\u5217\u5f0f\u7684\u6b63\u8d1f\u53f7\uff1b<\/li>\n
  5. \u4e24\u884c(\u5217)\u6210\u6bd4\u4f8b\uff0c\u884c\u5217\u5f0f\u503c\u4e3a0\uff1b<\/li>\n
  6. \u5c06\u4e00\u884c(\u5217)\u7684$k$\u500d\u52a0\u5230<\/strong>\u53e6\u4e00\u884c(\u5217)\uff0c\u503c\u4e0d\u53d8\uff1b<\/li>\n
  7. \u884c\u5217\u5f0f\u201c\u8f6c\u7f6e\u201d\uff0c\u503c\u4e0d\u53d8\uff1b<\/li>\n<\/ol>\n

    1.2 \u62c9\u666e\u62c9\u65af\u5c55\u5f00<\/h3>\n

      \u8bbe$\\rm B$\u662f\u4e00\u4e2a$n-{\\rm by}-n$\u7684\u77e9\u9635\u3002$\\rm B$\u7684\u7b2c$i$\u884c$j$\u5217\u7684\u4f59\u5b50\u5f0f<\/strong>${\\rm M}_{ij}$\u6307$\\rm B$\u4e2d\u53bb\u6389$i$\u884c$j$\u5217\u7684$n-1$\u9636\u5b50\u77e9\u9635\u7684\u884c\u5217\u5f0f\u3002${\\rm M}_{ij}$\u7684\u4ee3\u6570\u4f59\u5b50\u5f0f<\/strong>\u8868\u793a\u4e3a${\\rm C}_{ij} = (-i)^{i+j}{\\rm M}_{ij}$\u3002<\/p>\n

      \u884c\u5217\u5f0f$\\left| {\\rm B} \\right|$\uff0c\u6cbf$i$\u884c\u6216$j$\u5217\u5c55\u5f00\u5982\u4e0b\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n\\left| {\\rm B} \\right| &=&
    \n{\\rm b}_{i1}{\\rm C}_{i1} + {\\rm b}_{i2}{\\rm C}_{i2} + \\dots + {\\rm b}_{in}{\\rm C}_{in} \\newline &=&
    \n{\\rm b}_{1j}{\\rm C}_{1j} + {\\rm b}_{2j}{\\rm C}_{2j} + \\dots + {\\rm b}_{nj}{\\rm C}_{nj}
    \n\\end{eqnarray}
    \n$$<\/p>\n

      \u6211\u4eec\u6765\u770b\u4e00\u4e2a\u6570\u503c\u5316\u7684\u4f8b\u5b50\uff0c\u8003\u8651\u5982\u4e0b\u77e9\u9635\uff1a<\/p>\n

    $$
    \n{\\rm B}=
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{1} & {2} & {3} \\newline
    \n{4} & {5} & {6} \\newline
    \n{7} & {8} & {9}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u6211\u4eec\u53ef\u4ee5\u6cbf\u7740\u7b2c1\u884c\u5c55\u5f00\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n\\left| {\\rm B} \\right| &=&
    \n1 \\cdot
    \n\\left|
    \n\\begin{array}{*{20}{c}}
    \n{5} & {6} \\newline
    \n{8} & {9}
    \n\\end{array}
    \n\\right|
    \n-2 \\cdot
    \n\\left|
    \n\\begin{array}{*{20}{c}}
    \n{4} & {6} \\newline
    \n{7} & {9}
    \n\\end{array}
    \n\\right|
    \n+3 \\cdot
    \n\\left|
    \n\\begin{array}{*{20}{c}}
    \n{4} & {5} \\newline
    \n{7} & {8}
    \n\\end{array}
    \n\\right| \\newline &=&
    \n1 \\cdot (-3) -2 \\cdot (-6) +3 \\cdot (-3) \\newline &=&
    \n0
    \n\\end{eqnarray}
    \n$$<\/p>\n

    \u6211\u4eec\u4e5f\u53ef\u4ee5\u6cbf\u7740\u7b2c2\u5217\u5c55\u5f00\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n\\left| {\\rm B} \\right| &=&
    \n-2 \\cdot
    \n\\left|
    \n\\begin{array}{*{20}{c}}
    \n{4} & {6} \\newline
    \n{7} & {9}
    \n\\end{array}
    \n\\right|
    \n+5 \\cdot
    \n\\left|
    \n\\begin{array}{*{20}{c}}
    \n{1} & {3} \\newline
    \n{7} & {9}
    \n\\end{array}
    \n\\right|
    \n-8 \\cdot
    \n\\left|
    \n\\begin{array}{*{20}{c}}
    \n{1} & {3} \\newline
    \n{4} & {6}
    \n\\end{array}
    \n\\right| \\newline &=&
    \n-2 \\cdot (-6) +5 \\cdot (-12) -8 \\cdot (-6) \\newline &=&
    \n0
    \n\\end{eqnarray}
    \n$$<\/p>\n

    1.3 \u83b1\u5e03\u5c3c\u5179\u516c\u5f0f<\/h3>\n

      \u53c2\u8003\u300aLinear Algebra 01 Matrix\u300b\u4e2d\u7684\u7b2c\u516d\u7ae0\u201c\u9006\u77e9\u9635\u201d\u4e2d\u7b80\u5355\u8ba8\u8bba\u3002<\/p>\n

    \u4e8c\u3001\u7279\u5f81\u503c<\/h2>\n

    2.1 \u6982\u5ff5<\/h3>\n

      \u8ba9$\\rm A$\u4e3a\u4e00\u4e2a\u65b9\u9635\uff0c$\\rm x$\u662f\u4e00\u4e2a\u5217\u5411\u91cf\uff0c$\\lambda$\u662f\u6807\u91cf\uff0c\u5947\u5f02\u503c\u95ee\u9898\u7531\u4e0b\u7ed9\u51fa\uff1a<\/p>\n

    $$
    \n{\\rm Ax} = \\lambda {\\rm m}
    \n$$<\/p>\n

    \u5bf9\u4e8e\u4e0a\u8ff0\u65b9\u7a0b\uff0c\u6211\u4eec\u5c06\u5176\u6539\u5199\u4e3a\u9f50\u6b21\u65b9\u7a0b\u7684\u5f62\u5f0f\uff1a<\/p>\n

    $$
    \n({\\rm A} - \\lambda{\\rm I}){\\rm x} = 0
    \n$$<\/p>\n

    \u5176\u4e2d${\\rm A} - \\lambda{\\rm I}$\u5c31\u662f$\\rm A$\u7684\u5bf9\u89d2\u7ebf\u51cf\u53bb$\\lambda$\u3002\u4e3a\u4e86\u5b58\u5728\u975e\u96f6\u7279\u5f81\u5411\u91cf\uff0c\u77e9\u9635${\\rm A} - \\lambda{\\rm I}$\u5fc5\u987b\u662f\u5947\u5f02\u7684\uff0c\u5373\uff1a<\/p>\n

    $$
    \n{\\rm det}({\\rm A} - {\\lambda}{\\rm I}) = 0
    \n$$<\/p>\n

    \u8fd9\u4e2a\u7b49\u5f0f\u88ab\u79f0\u4e3a\u77e9\u9635$\\rm A$\u7684\u7279\u5f81\u65b9\u7a0b(characteristic equation)<\/strong>\uff0c\u7531\u83b1\u5e03\u5c3c\u5179\u65b9\u7a0b\u53ef\u4ee5\u77e5\u9053\uff0c\u7279\u5f81\u65b9\u7a0b\u662f\u4e00\u4e2a$n-{\\rm by}-n$\u7684\u77e9\u9635\u7684\u5173\u4e8e$\\lambda$\u7684$n$\u9636\u591a\u9879\u5f0f\u3002\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u627e\u5230\u7684\u7279\u5f81\u503c$\\lambda i$\u548c\u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf${\\rm x}_i$\uff0c\u53ef\u4ee5\u901a\u8fc7\u89e3\u5411\u91cf$\\rm x$\u7684\u65b9\u7a0b$({\\rm A} - {\\lambda}_i {\\rm I})x = 0$\u5f97\u5230\u3002<\/p>\n

      \u6211\u4eec\u901a\u8fc7\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u4f8b\u5b50\u6765\u8bf4\u660e\u8fd9\u4e2a\u95ee\u9898\uff1a<\/p>\n

    $$
    \n0 = {\\rm det}({\\rm A} - \\lambda {\\rm I}) = \\left|
    \n\\begin{array}{*{20}{c}}
    \n{a - \\lambda} & {b} \\newline
    \n{c} & {d - \\lambda}
    \n\\end{array}
    \n\\right| =
    \n(a - \\lambda)(d - \\lambda) - bc =
    \n{\\lambda}^{2} - (a+d){\\lambda} + (ad-bc)
    \n$$<\/p>\n

    \u8fd9\u4e2a\u7279\u5f81\u65b9\u7a0b\u53ef\u4ee5\u91cd\u5199\u4e3a\uff1a<\/p>\n

    $$
    \n{\\lambda}^{2} - {\\rm Tr(A)}\\lambda + {\\rm det(A)} = 0
    \n$$<\/p>\n

    \u5176\u4e2d${\\rm Tr(A)}$\u79f0\u4e3a\u77e9\u9635\u7684\u8ff9(trace)<\/strong>\uff0c\u5b83\u662f\u77e9\u9635\u7684\u4e3b\u5bf9\u89d2\u7ebf\u5143\u7d20\u4e4b\u548c\u3002<\/p>\n

      \u7531\u4e8e$2-{\\rm by}-2$\u7684\u7279\u5f81\u65b9\u7a0b\u662f\u4e8c\u6b21\u65b9\u7a0b\uff0c\u56e0\u6b64\u5b83\u5177\u6709\uff1a(1)\u4e24\u4e2a\u4e0d\u540c\u7684\u5b9e\u6839\uff1b(2)\u4e2a\u4e0d\u540c\u7684\u590d\u5171\u8f6d\u6839\uff1b(3)\u4e00\u4e2a\u9000\u5316\u5b9e\u6839\u3002\u66f4\u4e00\u822c\u5730\uff0c\u7279\u5f81\u503c\u53ef\u4ee5\u662f\u5b9e\u6570\u6216\u590d\u6570\uff0c\u5e76\u4e14$n-{\\rm by}-n$\u77e9\u9635\u53ef\u4ee5\u5177\u6709\u5c11\u4e8e$n$\u4e2a\u4e0d\u540c\u7684\u7279\u5f81\u503c\u3002<\/p>\n

    2.2 \u4f8b\u5b50<\/h3>\n

      \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u77e9\u9635$\\rm A$\uff0c\u6211\u4eec\u8981\u6c42\u5b83\u7684\u7279\u5f81\u503c\u3002\u77e9\u9635$\\rm A$\u5b9a\u4e49\u5982\u4e0b\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{0} & {1} \\newline
    \n{1} & {0}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u6211\u4eec\u901a\u8fc7\u6c42\u5b83\u5173\u4e8e$\\rm x$\u7684\u7279\u5f81\u65b9\u7a0b${\\rm Ax} = \\lambda {\\rm x}$\u6765\u5f97\u5230$\\lambda$\uff0c\u6211\u4eec\u5c06\u5176\u6539\u5199\u4e3a\uff1a$({\\rm A} - {\\lambda}{\\rm I}){\\rm x} = 0$\uff0c\u4e8e\u662f\u6709\uff1a<\/p>\n

    $$
    \n{\\rm det}({\\rm A} - {\\lambda}{\\rm I}) = 0 = \\left|
    \n\\begin{array}{*{20}{r}}
    \n{- \\lambda} & {1} \\newline
    \n{1} & {- \\lambda}
    \n\\end{array}
    \n\\right| =
    \n{\\lambda}^2 - 1
    \n$$<\/p>\n

    \u5f97\u5230${\\lambda}_1 = 1, {\\lambda}_2 = -1$\u3002\u7b2c\u4e00\u4e2a\u7279\u5f81\u5411\u91cf\u901a\u8fc7$({\\rm A} - {\\lambda}_{1}{\\rm I}){\\rm x}$\u5f97\u5230\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{-1} & {1} \\newline
    \n{1} & {-1}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{x_1} \\newline
    \n{x_2}
    \n\\end{array}
    \n\\right) = 0
    \n$$<\/p>\n

    \u5f97\u5230$x_1 = x_2$\u3002\u540c\u7406\uff0c\u7b2c\u4e8c\u4e2a\u7279\u5f81\u5411\u91cf\u901a\u8fc7$({\\rm A} - {\\lambda}_{2}{\\rm I}){\\rm x}$\u5f97\u5230\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{1} & {1} \\newline
    \n{1} & {1}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{x_1} \\newline
    \n{x_2}
    \n\\end{array}
    \n\\right) = 0
    \n$$<\/p>\n

    \u5f97\u5230$x_1 = -x_2$\u3002\u6211\u4eec\u5c06\u7ed3\u679c\u5199\u4e3a\uff1a<\/p>\n

    $$
    \n{\\lambda}_{1} = 1, {x}_{1} =
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{1} \\newline
    \n{1}
    \n\\end{array}
    \n\\right); \\quad
    \n{\\lambda}_{2} = -1, {x}_{2} =
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{1} \\newline
    \n{-1}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u5176\u4e2d$x_1$\u548c$x_2$\u4ecd\u9009\u4e00\u4e2a\u4f5c\u4e3a\u5e38\u91cf\uff0c\u53e6\u4e00\u4e2a\u4f5c\u4e3a\u53d8\u91cf\u5373\u53ef\u3002<\/p>\n

      \u8fd9\u91cc\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u5bf9\u4e8e\u4efb\u610fn\u9636\u65b9\u9635\u90fd\u6709\uff1a${\\lambda}_{1} + {\\lambda}_{2} = {\\rm Tr(A)}$\uff0c\u5e76\u4e14${\\lambda}_{1}{\\lambda}_{2}={\\rm det(A)}$\u3002<\/p>\n

    \u4e09\u3001\u77e9\u9635\u5bf9\u89d2\u5316<\/h2>\n

    3.1 \u6982\u5ff5<\/h3>\n

      \u6211\u4eec\u4ece\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u77e9\u9635$\\rm A$\u6765\u8003\u5bdf\uff0c\u5176\u7279\u5f81\u503c\u5982\u4e0b\u7ed9\u51fa\uff1a<\/p>\n

    $$
    \n{\\lambda}_{1}, {x}_{1} =
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{{x}_{11}} \\newline
    \n{{x}_{21}}
    \n\\end{array}
    \n\\right); \\quad
    \n{\\lambda}_{2}, {x}_{2} =
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{{x}_{12}} \\newline
    \n{{x}_{22}}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u540c\u65f6\u6211\u4eec\u8003\u8651\u5bf9\u77e9\u9635\u505a\u5982\u4e0b\u7684\u56e0\u5f0f\u5206\u89e3\uff1a<\/p>\n

    $$
    \n{\\rm A} \\left(
    \n\\begin{array}{*{20}{r}}
    \n{{x}_{11}} & {{x}_{12}} \\newline
    \n{{x}_{21}} & {{x}_{22}}
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{{\\lambda}_{1}{x}_{11}} & {{\\lambda}_{2}{x}_{12}} \\newline
    \n{{\\lambda}_{1}{x}_{21}} & {{\\lambda}_{2}{x}_{22}}
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{{x}_{11}} & {{x}_{12}} \\newline
    \n{{x}_{21}} & {{x}_{22}}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{{\\lambda}_{1}} & {0} \\newline
    \n{0} & {{\\lambda}_{2}}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u6211\u4eec\u5c06$\\rm S$\u5b9a\u4e49\u4e3a\u77e9\u9635$\\rm A$\u7684\u7279\u5f81\u5411\u91cf\u77e9\u9635\uff0c\u628a\u77e9\u9635$\\rm \\Lambda$\u5b9a\u4e49\u4e3a\u7279\u5f81\u503c\u7684\u5bf9\u89d2\u77e9\u9635\u3002\u90a3\u4e48\u5bf9\u4e8e\u4e00\u4e2an\u9636\u3001\u5e76\u4e14\u6709n\u4e2a\u7ebf\u6027\u65e0\u5173\u7684\u7279\u5f81\u5411\u91cf\u7684\u65b9\u9635\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n

    $$
    \n\\rm AS = S{\\Lambda}
    \n$$<\/p>\n

    \u5176\u4e2d$\\rm S$\u662f\u53ef\u9006\u77e9\u9635\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n

    $$
    \n\\rm A=S{\\Lambda}S^{-1}, \\quad {\\Lambda}={S}^{-1}AS
    \n$$<\/p>\n

    3.2 \u4f8b\u5b50<\/h3>\n

      \u6211\u4eec\u5c1d\u8bd5\u5bf9\u4ee5\u4e0b\u77e9\u9635\u8fdb\u884c\u5bf9\u89d2\u5316\uff1a<\/p>\n

    $$
    \n{\\rm A} \\left(
    \n\\begin{array}{*{20}{c}}
    \n{a} & {b} \\newline
    \n{b} & {a}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

      $\\rm A$\u7684\u7279\u5f81\u503c\u7531\u4ee5\u4e0b\u7ed9\u51fa\uff1a<\/p>\n

    $$
    \n{\\rm det}({\\rm A} - {\\lambda}{\\rm I}) = 0 = \\left|
    \n\\begin{array}{*{20}{c}}
    \n{a - {\\lambda}} & {b} \\newline
    \n{b} & {a - {\\lambda}}
    \n\\end{array}
    \n\\right| = (a - {\\lambda})^2 - b^2 = 0
    \n$$<\/p>\n

    \u4e8e\u662f${\\lambda}_{1} = a+b$\u3001${\\lambda}_{2} = a-b$\u3002\u5bf9\u4e8e${\\lambda}_{1}$\u6709\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{-b} & {b} \\newline
    \n{b} & {-b}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{{x}_{11}} \\newline
    \n{{x}_{21}}
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{0} \\newline
    \n{0}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u5bf9\u4e8e${\\lambda}_{2}$\u6709\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{b} & {b} \\newline
    \n{b} & {b}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{{x}_{12}} \\newline
    \n{{x}_{22}}
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{0} \\newline
    \n{0}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u4e8e\u662f\uff1a<\/p>\n

    $$
    \n{\\rm x_1} =
    \n\\frac{1}{\\sqrt{2}}
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{1} \\newline
    \n{1}
    \n\\end{array}
    \n\\right), \\quad
    \n{\\rm x_2} =
    \n\\frac{1}{\\sqrt{2}}
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{1} \\newline
    \n{-1}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u56e0\u4e3a\u7279\u5f81\u5411\u91cf$\\rm S$\u662f\u6b63\u4ea4\u7684\uff0c\u56e0\u6b64$\\rm S^{-1}= S^{T}$\uff0c\u6211\u4eec\u6709\uff1a<\/p>\n

    $$
    \n\\rm S = \\frac{1}{\\sqrt{2}}
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{1} & {1} \\newline
    \n{1} & {-1}
    \n\\end{array}
    \n\\right), \\quad
    \nS^{-1} = S^T = S
    \n$$<\/p>\n

    \u77e9\u9635\u5bf9\u89d2\u5316\u7684\u7ed3\u679c\u662f\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{a+b} & {0} \\newline
    \n{0} & {a-b}
    \n\\end{array}
    \n\\right)=
    \n\\frac{1}{2}
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{1} & {1} \\newline
    \n{1} & {-1}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{a} & {b} \\newline
    \n{b} & {a}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{r}}
    \n{1} & {1} \\newline
    \n{1} & {-1}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u56db\u3001\u77e9\u9635\u7684\u5e42<\/h2>\n

      \u77e9\u9635\u5bf9\u89d2\u5316\u6709\u52a9\u4e8e\u8ba1\u7b97\u77e9\u9635\u7684\u5e42\u3002\u5047\u8bbe\u77e9\u9635$\\rm A$\u53ef\u5bf9\u89d2\u5316\uff0c\u4e8e\u662f\uff1a<\/p>\n

    $$
    \n\\rm A^2 = (S{\\Lambda}S^{-1})(S{\\Lambda}S^{-1}) = S{\\Lambda}^{2}S^{-1}
    \n$$<\/p>\n

    \u5bf9\u4e8e$p$\u6b21\u5e42\u5219\u6709\uff1a<\/p>\n

    $$
    \n\\rm A^{p} = S{\\Lambda}^{p}S^{-1}
    \n$$<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u4e00\u3001\u884c\u5217\u5f0f   \u884c\u5217\u5f0f\u7684\u57fa\u672c\u5b9a\u4e49\u5df2\u7ecf\u5728\u300aLinear Algebra 01 Matrix\u300b\u4e2d\u7684\u7b2c\u516d\u7ae0\u201c\u9006\u77e9\u9635\u201d …<\/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\/1375"}],"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=1375"}],"version-history":[{"count":2,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1375\/revisions"}],"predecessor-version":[{"id":1377,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1375\/revisions\/1377"}],"wp:attachment":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/media?parent=1375"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/categories?post=1375"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/tags?post=1375"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}