{"id":1358,"date":"2024-05-30T16:11:21","date_gmt":"2024-05-30T08:11:21","guid":{"rendered":"https:\/\/swordofmorning.com\/?p=1358"},"modified":"2025-10-09T13:55:02","modified_gmt":"2025-10-09T05:55:02","slug":"linear-algebra-01","status":"publish","type":"post","link":"https:\/\/swordofmorning.com\/index.php\/2024\/05\/30\/linear-algebra-01\/","title":{"rendered":"\u7ebf\u6027\u4ee3\u6570\uff1a\u77e9\u9635"},"content":{"rendered":"

<\/div><\/p>\n

\u4e00\u3001\u77e9\u9635\u5b9a\u4e49<\/h2>\n

  \u4e00\u4e2a$m-{\\rm by}-n$\u7684\u77e9\u9635\u662f\u6307\u4e00\u4e2a\u5177\u6709$n$\u884c\u3001$m$\u5217\u7684\uff0c\u5305\u542b\u4e86\u6570\u5b57\u6216\u8005\u5176\u4ed6\u6570\u5b66\u5bf9\u8c61\u7684\u77e9\u5f62\u3002\u6bd4\u5982\uff0c\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u77e9\u9635$\\rm A$\uff0c\u5b83\u6709\u77402\u884c2\u5217\uff1a<\/p>\n

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

\u5176\u4e2d\u7b2c1\u884c\u6709$a$\u548c$b$\u4e24\u4e2a\u5143\u7d20\uff0c\u7b2c\u4e8c\u884c\u6709$c$\u548c$d$\u4e24\u4e2a\u5143\u7d20\u3002\u540c\u65f6\uff0c\u6211\u4eec\u53ef\u4ee5\u6784\u5efa\u4e00\u4e2a$2-{\\rm by}-3$\u7684\u77e9\u9635$\\rm B$\u548c\u4e00\u4e2a$3-{\\rm by}-2$\u7684\u77e9\u9635$\\rm C$\uff1a<\/p>\n

$$
\n{\\rm B} = \\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b}&{c} \\newline
\n{d}&{e}&{f}
\n\\end{array}
\n\\right)
\n,
\n{\\rm C} = \\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b} \\newline
\n{b}&{e} \\newline
\n{c}&{f}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u7279\u522b\u7684\uff0c\u5bf9\u4e8e$1-{\\rm by}-n$\u7684\u77e9\u9635\u548c$n-{\\rm by}-1$\u7684\u77e9\u9635\uff0c\u5206\u522b\u53eb\u505a\u884c\u77e9\u9635\u548c\u5217\u77e9\u9635\uff0c\u4e5f\u6210\u4e3a\u884c\u5411\u91cf\u548c\u5217\u5411\u91cf\u3002\u4e0b\u9762\u662f$n = 3$\u7684\u5217\u5411\u91cf\uff1a<\/p>\n

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

\u4e0b\u9762\u662f$n = 3$\u7684\u884c\u5411\u91cf\uff1a<\/p>\n

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

  \u4e00\u4e2a$m-{\\rm by}-n$\u7684\u77e9\u9635\u7684\u4e00\u822c\u8868\u8fbe\u5f62\u5f0f\u5982\u4e0b\uff1a<\/p>\n

$$
\n{\\rm A} = \\left(
\n\\begin{array}{*{20}{l}}
\n{a_{11}}&{a_{12}}&{\\cdots}&{a_{1n}} \\newline
\n{a_{21}}&{a_{22}}&{\\cdots}&{a_{2n}} \\newline
\n{\\vdots}&{\\vdots}&{\\ddots}&{\\vdots} \\newline
\n{a_{m1}}&{a_{m2}}&{\\cdots}&{a_{mn}}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u5176\u4e2d\u7b2c$i$\u884c$j$\u5217\u7684\u5143\u7d20\u88ab\u8868\u793a\u4e3a$a_{ij}$\u3002<\/p>\n

\u4e8c\u3001\u77e9\u9635\u7684\u52a0\u6cd5\u548c\u4e58\u6cd5<\/h2>\n

  \u77e9\u9635\u7684\u52a0\u6cd5\u5177\u6709\u70b9\u5bf9\u70b9<\/em>\u7684\u5f62\u5f0f\uff0c\u4e0b\u9762\u662f\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u4f8b\u5b50\uff1a<\/p>\n

$$
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b} \\newline
\n{c}&{d}
\n\\end{array}
\n\\right)+
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{e}&{f} \\newline
\n{g}&{h}
\n\\end{array}
\n\\right)=
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{a+e}&{b+f} \\newline
\n{c+g}&{d+h}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

  \u5bf9\u4e8e\u77e9\u9635\u4e0e\u6807\u91cf\u4e58\u6cd5\uff0c\u4ecd\u7136\u9075\u5faa\u70b9\u5bf9\u70b9<\/em>\u7684\u5f62\u5f0f\uff1a<\/p>\n

$$
\nk \\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b} \\newline
\n{c}&{d}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{ka}&{kb} \\newline
\n{kc}&{kd}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

  \u5bf9\u4e8e\u77e9\u9635\u4e0e\u77e9\u9635\u4e4b\u95f4\u7684\u4e58\u6cd5\u4e00\u822c\u5206\u4e3across product\u548cdot product\uff0c\u540e\u8005\u662f\u70b9\u5bf9\u70b9<\/em>\u7684\uff0c\u800c\u524d\u8005\u5219\u4e0d\u662f\u3002\u6211\u4eec\u901a\u8fc7\u4e00\u4e2a$2-{\\rm by}-2$\u77e9\u9635\u7684\u4f8b\u5b50\u6765\u8bf4\u660e\uff1a<\/p>\n

$$
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b} \\newline
\n{c}&{d}
\n\\end{array}
\n\\right) \\times
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{e}&{f} \\newline
\n{g}&{h}
\n\\end{array}
\n\\right)=
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{ae+bg} & {ef+bh} \\newline
\n{ce+dg} & {cf+dh}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u6211\u4eec\u53d1\u73b0\uff1a\u5bf9\u4e8e\u5143\u7d20(1,1)\u6765\u8bf4\uff0c\u5176\u7ed3\u679c\u662f\uff0c\u77e9\u9635\u7684\u7b2c1\u884c$\\cdot$\u77e9\u9635\u7684\u7b2c1\u5217\uff0c\u5e76\u6c42\u548c\uff1b\u5bf9\u4e8e\u5143\u7d20(1,2)\u6765\u8bf4\uff0c\u5176\u7ed3\u679c\u662f\u77e9\u9635\u7684\u7b2c1\u884c$\\cdot$\u77e9\u9635\u7684\u7b2c2\u5217\uff0c\u5e76\u6c42\u548c\u3002\u56e0\u6b64\u6211\u4eec\u4e0d\u96be\u53d1\u73b0\uff0c\u548c\u6807\u91cf\u7684\u4e58\u6cd5\u4e0d\u540c\uff0c\u77e9\u9635\u4e4b\u95f4\u7684\u53c9\u4e58\u4e0d\u662f<\/strong>\u53ef\u4ea4\u6362\u7684(commutative)\u3002\u6211\u4eec\u4ea4\u6362\u4e0a\u8ff0\u4e24\u4e2a\u77e9\u9635\uff1a<\/p>\n

$$
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{e}&{f} \\newline
\n{g}&{h}
\n\\end{array}
\n\\right) \\times
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b} \\newline
\n{c}&{d}
\n\\end{array}
\n\\right)=
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{ea+fc} & {eb+fd} \\newline
\n{ga+hc} & {gb+hd}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

  \u6211\u4eec\u603b\u7ed3\u4e0a\u9762\u7684\u4e58\u6cd5\uff0c\u5e76\u5c06\u5176\u6269\u5c55\u5230$m-{\\rm by}-n$\u7684\u4e00\u822c\u60c5\u51b5\u3002\u7ed3\u679c\u77e9\u9635\u7684\u7b2c(i,j)\u5143\u7d20\u7684\u503c\uff0c\u7b49\u4e8e$\\sum{(\\text{\u77e9\u9635\u7684i\u884c} \\cdot \\text{\u77e9\u9635\u7684j\u5217})}$\u3002\u4e0d\u96be\u53d1\u73b0\uff0ci\u884c\u7684\u5143\u7d20\u4e2a\u6570\u5e94\u8be5\u548cj\u5217\u7684\u5143\u7d20\u4e2a\u6570\u76f8\u7b49\uff0c\u5373\uff0c\u524d\u4e00\u4e2a\u77e9\u9635\u7684\u5217\u5e94\u8be5\u548c\u540e\u4e00\u4e2a\u77e9\u9635\u7684\u884c\u76f8\u7b49\u3002<\/p>\n

  \u5982\u679c\u6211\u4eec\u7528$\\rm A$\u6765\u8868\u793a\u4e00\u4e2a$m-{\\rm by}-p$\u7684\u77e9\u9635\uff0c\u5176\u5143\u7d20\u7528$a_{ij}$\u6765\u8868\u793a\uff1b\u7528$\\rm B$\u6765\u8868\u793a\u4e00\u4e2a$p-{\\rm by}-n$\u7684\u77e9\u9635\uff0c\u5176\u5143\u7d20\u7528$b_{ij}$\u6765\u8868\u793a\u3002\u90a3\u4e48\uff0c$\\rm C=AB$\u5c31\u662f\u4e00\u4e2a$m-{\\rm by}-p$\u7684\u77e9\u9635\uff0c\u5176\u4e2d\u7684\u5143\u7d20$c_{ij}$\u53ef\u4ee5\u8868\u793a\u4e3a\uff1a<\/p>\n

$$
\nC_{ij} = \\mathop{\\sum} \\limits_{k=1}^{n} {a_{ik} \\cdot b_{kj}}
\n$$<\/p>\n

\u4e09\u3001\u7279\u6b8a\u77e9\u9635<\/h2>\n

  \u5bf9\u4e8e\u96f6\u77e9\u9635(zero matrix)<\/strong>\u6765\u8bf4\uff0c\u5b83\u76f4\u63a5\u5199\u4f5c$0$\uff0c\u5b83\u5145\u5f53\u6807\u91cf\u4e2d0\u7684\u4f5c\u7528\uff0c\u56e0\u6b64\u5176\u5927\u5c0f\u53ef\u4ee5\u662f\u4efb\u610f\u7684\u3002\u5355\u4f4d\u77e9\u9635(identify matrix)<\/strong>\u662f\u4e00\u4e2a\u65b9\u9635\uff0c\u5b83\u7684\u4e3b\u5bf9\u89d2\u7ebf\u4e0a\u7684\u503c\u5747\u4e3a1\uff0c\u8bb0\u4f5c$\\rm I$\u3002\u5355\u4f4d\u77e9\u9635\u5145\u5f53\u6807\u91cf\u4e2d1\u7684\u4f5c\u7528\uff0c\u5982\u679c$\\rm A$\u662f\u4e00\u4e2a\u548c$\\rm I$\u540c\u5f62\u7684\u77e9\u9635\uff0c\u90a3\u4e48\u6709\uff1a<\/p>\n

$$
\n\\rm AI = A = IA
\n$$<\/p>\n

\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u96f6\u77e9\u9635\u548c\u5355\u4f4d\u77e9\u9635\u5982\u4e0b\uff1a<\/p>\n

$$
\n0 = \\left(
\n\\begin{array}{*{20}{l}}
\n{0}&{0} \\newline
\n{0}&{0}
\n\\end{array}
\n\\right),{\\rm I} = \\left(
\n\\begin{array}{*{20}{l}}
\n{1}&{0} \\newline
\n{0}&{1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

  \u5bf9\u89d2\u77e9\u9635(diagonal matrix)<\/strong>\u4ec5\u5728\u5bf9\u89d2\u7ebf\u4e0a\u6709\u975e\u96f6\u5143\u7d20\uff0c\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u5bf9\u89d2\u77e9\u9635\u5982\u4e0b\uff1a<\/p>\n

$$
\n{\\rm D} = \\left(
\n\\begin{array}{*{20}{l}}
\n{d_{1}}&{0} \\newline
\n{0}&{d_{2}}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u901a\u5e38\uff0c\u5bf9\u89d2\u77e9\u9635\u662f\u4e00\u4e2a\u65b9\u9635\u3002\u4f46\u5982\u679c\u662f\u4e00\u4e2a\u6ee1\u8db3$d_{ij}=0, i \\neq j$\u7684\u77e9\u5f62\u77e9\u9635\uff0c\u4e5f\u53ef\u4ee5\u88ab\u8ba4\u4e3a\u662f\u5bf9\u89d2\u77e9\u9635\u3002<\/p>\n

  \u5e26\u72b6\u77e9\u9635(band or banded matrix)<\/strong>\uff0c\u76f8\u5f53\u4e8e\u5bf9\u89d2\u77e9\u9635\u7684\u6269\u5c55\u7248\uff0c\u5b83\u8981\u6c42\u4ec5\u5728\u5bf9\u89d2\u5e26\u4e0a\u5b58\u5728\u975e\u96f6\u5143\u7d20\uff0c\u4f8b\u5982\u4e00\u4e2a$3-{\\rm by}-3$\u7684\u5e26\u72b6\u77e9\u9635\uff1a<\/p>\n

$$
\n{\\rm B} = \\left(
\n\\begin{array}{*{20}{l}}
\n{d_{1}}&{a_{1}}&{0} \\newline
\n{b_{1}}&{d_{2}}&{a_{2}} \\newline
\n{0}&{b_{2}}&{d_{3}}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

  \u4e0a\u4e09\u89d2\u77e9\u9635(upper)<\/strong>\u6216\u4e0b\u4e09\u89d2\u77e9\u9635(lower triangular matrix )<\/strong>\uff0c\u662f\u4e00\u4e2a\u65b9\u9635\uff0c\u4ec5\u5728\u5bf9\u89d2\u7ebf\u548c\u5176\u4e0a\u65b9\u3001\u4e0b\u65b9\u5b58\u5728\u975e\u96f6\u5143\u7d20\uff0c\u4f8b\u5982\u4e00\u4e2a$3-{\\rm by}-3$\u7684\u4e0a\u4e09\u89d2\u548c\u4e0b\u4e09\u89d2\u77e9\u9635\uff1a<\/p>\n

$$
\n{\\rm U} = \\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{d}&{f} \\newline
\n{0}&{b}&{e} \\newline
\n{0}&{0}&{c}
\n\\end{array}
\n\\right),{\\rm L} = \\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{0}&{0} \\newline
\n{d}&{b}&{0} \\newline
\n{f}&{e}&{c}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u56db\u3001\u8f6c\u7f6e<\/h2>\n

  \u5bf9\u4e8e\u77e9\u9635$\\rm A$\uff0c\u5176\u8f6c\u7f6e\u77e9\u9635(transpose matrix)<\/strong>\u5199\u4f5c$\\rm A^{T}$\uff0c\u5bf9\u4e8e\u4e00\u4e2a$m-{\\rm by}-n$\u7684\u77e9\u9635$\\rm A$\u6709\uff1a<\/p>\n

$$
\n{\\rm A} = \\left(
\n\\begin{array}{*{20}{l}}
\n{a_{11}}&{a_{12}}&{\\cdots}&{a_{1n}} \\newline
\n{a_{21}}&{a_{22}}&{\\cdots}&{a_{2n}} \\newline
\n{\\vdots}&{\\vdots}&{\\ddots}&{\\vdots} \\newline
\n{a_{m1}}&{a_{m2}}&{\\cdots}&{a_{mn}}
\n\\end{array}
\n\\right)
\n,
\n{\\rm A^{T}} = \\left(
\n\\begin{array}{*{20}{l}}
\n{a_{11}}&{a_{21}}&{\\cdots}&{a_{m1}} \\newline
\n{a_{12}}&{a_{22}}&{\\cdots}&{a_{m2}} \\newline
\n{\\vdots}&{\\vdots}&{\\ddots}&{\\vdots} \\newline
\n{a_{1n}}&{a_{2n}}&{\\cdots}&{a_{mn}}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u6362\u8a00\u4e4b\uff0c\u6211\u4eec\u53ef\u4ee5\u5199\u4f5c\uff1a<\/p>\n

$$
\na_{ij}^{\\rm T} = a_{ji}
\n$$<\/p>\n

  \u663e\u7136\uff0c\u5982\u679c\u4e00\u4e2a\u77e9\u9635\u5927\u5c0f\u662f$m-{\\rm by}-n$\u7684\uff0c\u90a3\u4e48\u5176\u8f6c\u7f6e\u77e9\u9635\u7684\u5927\u5c0f\u5219\u662f$n-{\\rm by}-m$\u7684\u3002\u6211\u4eec\u4ee5\u4e00\u4e2a$3-{\\rm by}-2$\u7684\u77e9\u9635\u4e3a\u4f8b\uff1a<\/p>\n

$$
\n{\\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{d} \\newline
\n{b}&{e} \\newline
\n{c}&{f}
\n\\end{array}
\n\\right)}^{\\rm T}=
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b}&{c} \\newline
\n{d}&{e}&{f}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

  \u4e0b\u9762\u662f\u4e24\u4e2a\u5bb9\u6613\u8bc1\u660e\u7684\u4e8b\u5b9e\uff1a<\/p>\n

$$
\n\\rm ({A}^{T})^{T} = {A}, (A+B)^{T} = A^{T}+B^{T}
\n$$<\/p>\n

\u800c\u548c\u6807\u91cf\u4e0d\u540c\uff0c\u77e9\u9635\u4e58\u79ef\u7684\u8f6c\u7f6e\u7b49\u4e8e\u77e9\u9635\u8f6c\u7f6e\u7684\u4e58\u79ef\uff1a<\/p>\n

$$
\n\\rm (AB)^{T} = B^{T}A^{T}
\n$$<\/p>\n

\u5176\u8bc1\u660e\u5982\u4e0b\uff1a<\/p>\n

\u5bf9\u4e8e\u77e9\u9635$\\rm A$\u548c\u77e9\u9635$\\rm B$\u7684\u53c9\u4e58\u800c\u8a00\uff0c\u6709\uff1a<\/p>\n

$$
\n\\begin{eqnarray}
\n\\left\\{
\n\\begin{array}{**lr**}
\n({\\rm AB})_{ij} = row({\\rm A})_{i} \\cdot col({\\rm B}_{j}) \\newline
\n({\\rm AB})_{ij}^{\\rm T} = row({\\rm A})_{j} \\cdot col({\\rm B}_{i})
\n\\end{array}
\n\\right. \\newline
\n\\left\\{
\n\\begin{array}{**lr**}
\nrow({\\rm A})_{i} = col({\\rm A}^{T})_{i} \\newline
\ncol({\\rm B})_{j} = row({\\rm B}^{T})_{j}
\n\\end{array}
\n\\right.
\n\\end{eqnarray}
\n$$<\/p>\n

\u6211\u4eec\u5c06\u4e0b\u9762\u7684\u7b49\u5f0f\u5e26\u5165\u4e0a\u9762\u7684\u7b49\u5f0f\u4e2d\uff1a<\/p>\n

$$
\n\\begin{eqnarray}
\n({\\rm AB})_{ij} &=& row({\\rm A})_{i} \\cdot col({\\rm B}_{j}) \\newline
\n&=& col({\\rm A}^{T})_{i} \\cdot row({\\rm B}^{T})_{j} \\newline
\n&=& row({\\rm B}^{T})_{j} \\cdot col({\\rm A}^{T})_{i} \\newline
\n&=& {\\rm (B^{T} A^{T})}_{ji}
\n\\end{eqnarray}
\n$$<\/p>\n

\u6211\u4eec\u5c06\u5176\u7528\u77e9\u9635\u63cf\u8ff0\u5219\u6709\uff1a<\/p>\n

$$
\n\\begin{eqnarray}
\n({\\rm AB}) &=& {\\rm (B^{T} A^{T})}^{T} \\newline
\n({\\rm AB})^{\\rm T} &=& {\\rm (B^{T} A^{T})}
\n\\end{eqnarray}
\n$$<\/p>\n

\u8bc1\u6bd5\u3002<\/p>\n

  \u6b64\u5916\uff0c\u8f6c\u7f6e\u77e9\u9635\u4e2d\u4e5f\u5b58\u5728\u4e00\u4e9b\u7279\u6b8a\u7684\u77e9\u9635\u3002\u5bf9\u79f0\u77e9\u9635(symmetric matrix)<\/strong>\u6307\u7684\u662f$\\rm A^T = A$\uff1b\u800c\u659c\u5bf9\u79f0\u77e9\u9635(skew symmetric matrix)<\/strong>\u6307\u7684\u662f$\\rm A^T = -A$\u3002\u4e0b\u9762\u662f\u4e00\u4e2a$3-{\\rm by}-3$\u7684\u4f8b\u5b50\uff1a<\/p>\n

$$
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{a}&{b}&{c} \\newline
\n{b}&{d}&{e} \\newline
\n{c}&{e}&{f}
\n\\end{array}
\n\\right)
\n,
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{0}&{b}&{c} \\newline
\n{-b}&{0}&{e} \\newline
\n{-c}&{-e}&{0}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u8fd9\u91cc\u9700\u8981\u6ce8\u610f\u7684\u662f\uff0c\u659c\u5bf9\u79f0\u77e9\u9635\u7684\u5bf9\u89d2\u7ebf\u5fc5\u987b\u4e3a0\u3002<\/p>\n

\u4e94\u3001\u5185\u79ef\u548c\u5916\u79ef<\/h2>\n

  \u4e24\u4e2a\u5411\u91cf\u7684\u5185\u79ef(inner product, dot product, scalar product)<\/strong>\u4ece\u884c\u5411\u91cf\u548c\u5217\u5411\u91cf\u7684\u4e58\u79ef\u4e2d\u83b7\u5f97\u3002\u8fd9\u91cc\u6211\u4eec\u9ed8\u8ba4\u5411\u91cf\u90fd\u662f\u5217\u5411\u91cf\uff0c\u56e0\u6b64\u4e0b\u9762\u662f\u4e24\u4e2a$3-{\\rm by}-1$\u7684\u5217\u5411\u91cf$\\rm u$\u548c$\\rm v$\u76f8\u4e58\u5f97\u5230\u5185\u79ef\u7684\u4f8b\u5b50\uff1a<\/p>\n

$$
\n{\\rm {u}^{T}{v}} =
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{u_1}&{u_2}&{u_3}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{v_1} \\newline
\n{v_2} \\newline
\n{v_3}
\n\\end{array}
\n\\right)
\n= {u_1}{v_1} + {u_2}{v_2} + {u_3}{v_3}
\n$$<\/p>\n

\u5982\u679c\u4e24\u4e2a\u975e\u96f6\u5411\u91cf\u7684\u5185\u79ef\u662f\u96f6\uff0c\u90a3\u4e48\u6211\u4eec\u8bf4\u8fd9\u4e24\u4e2a\u5411\u91cf\u662f\u6b63\u4ea4\u7684(orthogonal)<\/strong>\u3002\u5411\u91cf\u7684\u8303\u6570(norm)<\/strong>\u88ab\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$
\n{\\left \\Vert {\\rm u} \\right \\Vert} =
\n{\\left( {\\rm {u}^{T}{u}} \\right)}^{1\/2}=
\n{\\left( {u_1^2 + u_2^2 + u_3^2} \\right)}^{1\/2}
\n$$<\/p>\n

\u5982\u679c\u5411\u91cf\u7684\u8303\u6570\u7b49\u4e8e\u4e00\uff0c\u6211\u4eec\u8bf4\u8fd9\u4e2a\u5411\u91cf\u5df2\u7ecf\u5f52\u4e00\u5316(normalized)<\/strong>\u3002\u5982\u679c\u4e00\u7ec4\u5411\u91cf\u5b83\u4eec\u76f8\u4e92\u6b63\u4ea4\u4e14\u662f\u5f52\u4e00\u7684\uff0c\u90a3\u4e48\u5b83\u4eec\u88ab\u79f0\u4e3a\u6807\u51c6\u6b63\u4ea4\u57fa(orthonormal basis)<\/strong>\u3002<\/p>\n

  \u5411\u91cf\u7684\u5916\u79ef\u5219\u662f\u88ab\u5b9a\u4e49\u4e3a\u5217\u5411\u91cf\u4e0e\u884c\u5411\u91cf\u7684\u4e58\u79ef\uff0c\u4e0b\u9762\u662f\u540c\u6837\u662f$\\rm u$\u548c$\\rm v$\u7684\u4f8b\u5b50\uff1a<\/p>\n

$$
\n{\\rm {u}{v}^{T}} =
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{u_1} \\newline
\n{u_2} \\newline
\n{u_3}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{v_1}&{v_2}&{v_3}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{{u}_{1}{v}_{1}} & {{u}_{1}{v}_{2}} & {{u}_{1}{v}_{3}} \\newline
\n{{u}_{2}{v}_{1}} & {{u}_{2}{v}_{2}} & {{u}_{2}{v}_{3}} \\newline
\n{{u}_{3}{v}_{1}} & {{u}_{3}{v}_{2}} & {{u}_{3}{v}_{3}}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

  \u77e9\u9635$\\rm B$\u7684\u8ff9(trace)<\/strong>\u88ab\u8bb0\u4f5c$\\rm Tr B$\uff0c\u662f$\\rm B$\u7684\u5bf9\u89d2\u5143\u7d20\u4e4b\u548c\u3002<\/p>\n

\u516d\u3001\u9006\u77e9\u9635<\/h2>\n

  \u65b9\u9635\u53ef\u80fd\u5b58\u5728\u9006(inverse)\u77e9\u9635<\/strong>\u3002\u5982\u679c\u77e9\u9635$\\rm A$\u5b58\u5728\u9006\u9635\uff0c\u6211\u4eec\u7528$\\rm A^{-1}$\u6765\u8868\u793a\u8fd9\u4e2a\u9006\u9635\u3002\u9006\u9635\u6ee1\u8db3\u5982\u4e0b\u6761\u4ef6\uff1a<\/p>\n

$$
\n\\rm {A}{A}^{-1} = I = {A}^{-1}{A}
\n$$<\/p>\n

  \u9996\u5148\uff0c\u6211\u4eec\u4ece\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u4f8b\u5b50\u4e2d\u6765\u770b\u5982\u4f55\u8ba1\u7b97\u77e9\u9635\u7684\u9006\u9635\uff1a<\/p>\n

$$
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{a} & {b} \\newline
\n{c} & {d}
\n\\end{array}
\n\\right)
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{x_1} & {x_2} \\newline
\n{x_3} & {x_4}
\n\\end{array}
\n\\right) =
\n\\left(
\n\\begin{array}{*{20}{l}}
\n{1} & {0} \\newline
\n{0} & {1}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u65b9\u7a0b\uff1a<\/p>\n

$$
\n\\begin{eqnarray}
\nax_1 + by_1 &=& 1 \\tag{6-1} \\newline
\nax_2 + by_2 &=& 0 \\tag{6-2} \\newline
\ncx_1 + dy_1 &=& 0 \\tag{6-3} \\newline
\ncx_2 + dy_2 &=& 1 \\tag{6-4}
\n\\end{eqnarray}
\n$$<\/p>\n

\u4e3a\u4e86\u6c42\u89e3$y_1$\u548c$y_2$\uff0c\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u5f0f(6-2)\u548c\u5f0f(6-3)\u7684\u9f50\u6b21\u65b9\u7a0b(homogenous equations)<\/strong>\u6765\u7528$x$\u8868\u793a$y$\uff1b\u4e3a\u4e86\u6c42\u89e3$x_1$\u548c$x_2$\uff0c\u6211\u4eec\u5219\u53ef\u4ee5\u5c06\u4e4b\u524d\u7b97\u51fa\u6765\u7684$y_1$\u548c$y_2$\u5e26\u5165\u5f0f(6-1)\u548c\u5f0f(6-4)\u7684\u975e\u9f50\u6b21\u65b9\u7a0b(inhomogeneous equations)<\/strong>\u4e2d\u3002\u4e8e\u662f\u6211\u4eec\u5f97\u5230\uff1a<\/p>\n

$$
\n\\begin{eqnarray}
\nx_1 &=& \\frac{d}{ad-bc} \\newline
\nx_2 &=& \\frac{-b}{ad-bc} \\newline
\ny_1 &=& \\frac{-c}{ad-bc} \\newline
\ny_2 &=& \\frac{a}{ad-bc}
\n\\end{eqnarray}
\n$$<\/p>\n

\u8fdb\u4e00\u6b65\uff1a<\/p>\n

$$
\n{\\left(
\n\\begin{array}{*{20}{l}}
\n{a} & {b} \\newline
\n{c} & {d}
\n\\end{array}
\n\\right)}^{-1} =
\n\\frac{1}{ad-bc}
\n\\left(
\n\\begin{array}{*{20}{c}}
\n{d} & {-b} \\newline
\n{-c} & {a}
\n\\end{array}
\n\\right)
\n$$<\/p>\n

\u5176\u4e2d$ad-bc$\u88ab\u79f0\u4e3a\u77e9\u9635\u7684\u884c\u5217\u5f0f(determinant)<\/strong>\uff0c\u77e9\u9635$\\rm A$\u7684\u884c\u5217\u5f0f\u8bb0\u4f5c$\\rm det(A)$\u6216\u8005$\\left| {\\rm A} \\right|$\u3002\u4e00\u4e2a$n$\u9636\u65b9\u9635$\\rm A$\u7684\u884c\u5217\u5f0f\u53ef\u4ee5\u88ab\u5b9a\u4e49\u4e3a\uff1a<\/p>\n

$$
\n{\\rm det(A)} =
\n\\mathop{\\sum} \\limits_{\\sigma \\in S_n}
\n{\\rm sgn}(\\sigma)
\n\\mathop{\\prod} \\limits_{i=1}^{n}
\na_{i,\\sigma(i)}
\n$$<\/p>\n

    \n
  1. $S_n$\u662f\u96c6\u5408\u4e0a\u7684\u7f6e\u6362\u5168\u4f53\u3002<\/li>\n
  2. $\\mathop{\\sum} \\limits_{\\sigma \\in S_n}$\u8868\u793a\u5bf9\u5168\u4f53$S_n$\u5143\u7d20\u6c42\u548c\uff0c\u5373\u5bf9\u4e8e\u6bcf\u4e2a${\\sigma \\in S_n}$\u800c\u8a00\uff0c${\\rm sgn}(\\sigma) \\mathop{\\prod} \\limits_{i=1}^{n}$\u53ea\u5728\u52a0\u6cd5\u7b97\u5f0f\u4e2d\u51fa\u73b0\u4e00\u6b21\uff1b\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u6ee1\u8db3$1 \\le i,j \\le n$\u7684\u6570\u5bf9$(i,j)$\uff0c$a_ij$\u662f\u77e9\u9635A\u7684\u7b2ci\u884c\u7b2cj\u5217\u5143\u7d20\u3002<\/li>\n
  3. ${\\rm sgn}(\\sigma)$\u8868\u793a\uff0c\u7f6e\u6362${\\sigma \\in S_n}$\u7684\u7b26\u53f7\u5dee\uff0c\u5177\u4f53\u5730\u8bf4\uff1a\u6ee1\u8db3$1 \\le i \\le j \\le n$\u4f46$\\sigma(i) \\gt \\sigma(j)$\u7684\u6709\u5e8f\u6570\u5bf9$(i,j)$\u88ab\u79f0\u4e3a$\\sigma$\u7684\u4e00\u4e2a\u9006\u5e8f\u3002\u5982\u679c$\\sigma$\u7684\u9006\u5e8f\u6570\u6709\u5076\u6570\u4e2a\uff0c\u5219${\\rm sgn}(\\sigma) = 1$\uff1b\u5982\u679c\u6709\u53ca\u6570\u4e2a\uff0c\u5219${\\rm sgn}(\\sigma) = -1$\u3002<\/li>\n<\/ol>\n

    \u4e3e\u4f8b\u6765\u8bf4\uff0c\u5bf9\u4e8e3\u5143\u7f6e\u6362$\\sigma = (2,3,1)$\uff0c\u5176\u9006\u5e8f\u6570\u4e3a$sum(0, 0, 2) = 2$\uff0c\u56e0\u6b64\u4e09\u9636\u884c\u5217\u5f0f\u4e2d\u51fa\u73b0\u7684$a_{1,2}a_{2,3}a_{3,1}$\u5176\u7b26\u53f7\u662f\u6b63\u7684\uff0c\u5373${{\\rm sgn}(\\sigma)} = 1$\uff1b\u4f46\u5bf9\u4e8e3\u5143\u7f6e\u6362$\\sigma = (3,2,1)$\u800c\u8a00\uff0c\u5176\u9006\u5e8f\u6570\u4e3a$sum(0, 1, 2) = 3$\uff0c\u56e0\u6b64\u4e09\u9636\u884c\u5217\u5f0f\u4e2d\u51fa\u73b0\u7684$a_{1,3}a_{2,2}a_{3,1}$\u5176\u7b26\u53f7\u662f\u8d1f\u7684\uff0c\u5373${{\\rm sgn}(\\sigma)} = -1$\u3002<\/p>\n

      \u548c\u77e9\u9635\u7684\u8f6c\u7f6e\u7c7b\u4f3c\uff0c\u6211\u4eec\u53ef\u4ee5\u8f7b\u6613\u5730\u5f97\u5230\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n\\rm {(AB)}^{-1} = {B}^{-1}{A}^{-1} \\tag{6-5} \\newline
    \n\\rm {(A^{T})}^{-1} = {(A^{-1})}^{-T} \\tag{6-6}
    \n\\end{eqnarray}
    \n$$<\/p>\n

    \u5f0f(6-5)\u8bc1\u660e\u5982\u4e0b\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n\\rm AB &=& \\rm C \\newline
    \n\\rm {A}^{-1}AB &=& \\rm {A}^{-1}C \\newline
    \n\\rm IB &=& \\rm {A}^{-1}C \\newline
    \n\\rm B &=& \\rm {A}^{-1}C \\newline
    \n\\rm {B}^{-1}B &=& \\rm {B}^{-1}{A}^{-1}C \\newline
    \n\\rm I &=& \\rm {B}^{-1}{A}^{-1}C \\newline
    \n\\rm {C}^{-1} &=& \\rm {B}^{-1}{A}^{-1}C{C}^{-1} \\newline
    \n\\rm {C}^{-1} &=& \\rm {B}^{-1}{A}^{-1} \\newline
    \n\\rm {(AB)}^{-1} &=& \\rm {B}^{-1}{A}^{-1}
    \n\\end{eqnarray}
    \n$$<\/p>\n

    \u5f0f(6-6)\u8bc1\u660e\u5982\u4e0b\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n\\rm {A}{A}^{-1} &=& \\rm I \\newline
    \n\\rm {({A}{A}^{-1})}^{T} &=& \\rm I^{T} \\newline
    \n\\rm {({A}^{-1})}^{T} {A}^{T} &=& \\rm I \\newline
    \n\\rm {({A}^{-1})}^{T} {A}^{T} {({A}^{T})}^{-1} &=& \\rm {({A}^{T})}^{-1} \\newline
    \n\\rm {({A}^{-1})}^{T} I &=& \\rm {({A}^{T})}^{-1} \\newline
    \n\\rm {({A}^{-1})}^{T} &=& \\rm {({A}^{T})}^{-1}
    \n\\end{eqnarray}
    \n$$<\/p>\n

    \u4e03\u3001\u6b63\u4ea4\u77e9\u9635<\/h2>\n

      \u5177\u6709\u5b9e\u6570\u9879\u7684\u65b9\u9635$\\rm Q$\u6ee1\u8db3\uff1a<\/p>\n

    $$
    \n{\\rm Q}^{-1} = {\\rm Q}^{\\rm T} \\tag{7-1}
    \n$$<\/p>\n

    \u8fd9\u4e2a\u77e9\u9635\u88ab\u79f0\u4e3a\u6b63\u4ea4\u77e9\u9635(orthogonal matrix)<\/strong>\uff0c\u5176\u53e6\u4e00\u4e2a\u5b9a\u4e49\u5199\u505a\uff1a<\/p>\n

    $$
    \n\\rm {Q}{Q}^{T} = I, {Q}^{T}{Q} = I \\tag{7-2}
    \n$$<\/p>\n

      \u4e0b\u9762\u662f\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u4f8b\u5b50\uff1a<\/p>\n

    $$
    \n{\\rm Q} =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{q_{11}} & {q_{12}} \\newline
    \n{q_{21}} & {q_{22}}
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{\\rm q_1} & {\\rm q_2} \\newline
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u5176\u4e2d$\\rm q_1$\u548c$\\rm q_2$\u662f\u4e00\u4e2a$2-{\\rm by}-1$\u7684\u5217\u5411\u91cf\uff0c\u4e8e\u662f\u6709\uff1a<\/p>\n

    $$
    \n\\rm {Q}^{T}{Q} =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{\\rm q_1^T} \\newline
    \n{\\rm q_1^T}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{\\rm q_1} & {\\rm q_2} \\newline
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{\\rm q_1^T q_1} & {\\rm q_1^T q_2} \\newline
    \n{\\rm q_2^T q_1} & {\\rm q_2^T q_2}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u5982\u679c$\\rm Q$\u662f\u6b63\u4ea4\u7684\uff0c\u90a3\u4e48\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n{\\rm q_1^T q_1} = {\\rm q_2^T q_2} = 1 \\newline
    \n{\\rm q_1^T q_2} = {\\rm q_2^T q_1} = 0
    \n\\end{eqnarray}
    \n$$<\/p>\n

    \u4e5f\u5c31\u662f\u8bf4\uff0c$\\rm Q$\u7684\u5217\u4e92\u76f8\u4e3a\u6b63\u4ea4\u5411\u91cf\u96c6\uff1b\u540c\u7406\uff0c\u4e5f\u9002\u7528\u4e8e\u5176\u884c\u5411\u91cf\u3002\u56e0\u6b64\uff0c\u6b63\u4ea4\u77e9\u9635\u7684\u7b49\u6548\u5b9a\u4e49\u4e5f\u53ef\u4ee5\u662f\uff1a\u4e00\u4e2a\u5177\u6709\u5b9e\u6570\u9879\u7684\u65b9\u9635\uff0c\u5176\u5217(\u4ee5\u53ca\u884c)\u662f\u4e00\u7ec4\u6b63\u4ea4\u5411\u91cf\u3002<\/p>\n

      \u6b63\u4ea4\u77e9\u9635\u7684\u7b2c\u4e09\u4e2a\u5b9a\u4e49\u5982\u4e0b\u3002\u53e6$\\rm Q$\u4e3a\u4e00\u4e2a$n-{\\rm by}-n$\u7684\u6b63\u4ea4\u77e9\u9635\uff0c\u53e6$\\rm x$\u4e3a\u4e00\u4e2a$n-{\\rm by}-1$\u7684\u5217\u5411\u91cf\u3002\u5411\u91cf$\\rm Qx$\u7684\u957f\u5ea6(\u8303\u6570\u3001\u6a21\u957f)\u7684\u5e73\u65b9\u53ef\u4ee5\u5199\u505a\uff1a<\/p>\n

    $$
    \n\\rm {\\left \\Vert{Qx} \\right \\Vert}^{2} = {(Qx)}^{T}(Qx) = {x}^{T}{Q}^{T}{Q}{x} = {x}^{T}{I}{x} = {x}^{T}{x} = {\\left \\Vert{x} \\right \\Vert}^{2} \\tag{7-3}
    \n$$<\/p>\n

    \u6211\u4eec\u53d1\u73b0$\\rm Qx$\u7684\u957f\u5ea6\u7b49\u4e8e$\\rm x$\u7684\u957f\u5ea6\uff0c\u56e0\u6b64\u6211\u4eec\u8bf4\u6b63\u4ea4\u77e9\u9635\u662f\u4fdd\u7559\u957f\u5ea6\u7684\u77e9\u9635\u3002\u5728\u4e0b\u4e00\u8282\u4e2d\uff0c\u6211\u4eec\u5c06\u770b\u5230\u6b63\u4ea4\u77e9\u9635\u7684\u4e00\u4e2a\u5e94\u7528\uff0c\u5b83\u5728\u4e8c\u7ef4\u7a7a\u95f4\u4e2d\u65cb\u8f6c\u4e00\u4e2a\u5411\u91cf\u3002<\/p>\n

    \u516b\u3001\u65cb\u8f6c\u77e9\u9635<\/h2>\n

      \u5047\u8bbe\u6211\u4eec\u73b0\u5728\u6709\u4e00\u4e2a\u4e8c\u7ef4\u5411\u91cf$(x,y)$\uff0c\u5176\u4e0e\u5750\u6807\u8f74\u7684\u5939\u89d2\u4e3a$\\phi$\uff0c\u5176\u957f\u5ea6\u4e3a$r$\uff1b\u65cb\u8f6c$\\theta$\u540e\u7684\u5411\u91cf\u4e3a$(x', y')$\u3002\u6211\u4eec\u7528\u4e09\u89d2\u51fd\u6570\u6765\u8868\u793a\u8fd9\u4e2a\u65b0\u7684\u5411\u91cf\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \nx' = r {\\rm cos}(\\theta + \\phi) = r ({\\rm cos}{\\theta}{\\rm cos}{\\phi} - {\\rm sin}{\\theta}{\\rm sin}{\\phi}) = x{\\rm cos}{\\theta} - y{\\rm sin}{\\theta} \\newline
    \ny' = r {\\rm sin}(\\theta + \\phi) = r ({\\rm sin}{\\theta}{\\rm cos}{\\phi} + {\\rm cos}{\\theta}{\\rm sin}{\\phi}) = x{\\rm sin}{\\theta} + y{\\rm cos}{\\theta}
    \n\\end{eqnarray}
    \n$$<\/p>\n

    \u5c06\u5176\u7528\u77e9\u9635\u7684\u5f62\u5f0f\u8868\u8fbe\u5219\u6709\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{x'} \\newline
    \n{y'}
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{{\\rm cos}{\\theta}} & {-{\\rm sin}{\\theta}} \\newline
    \n{{\\rm sin}{\\theta}} & {{\\rm cos}{\\theta}}
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{x} \\newline
    \n{y}
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u6211\u4eec\u7528$\\rm R_{\\theta}$\u6765\u8868\u793a\u4e0a\u9762\u8fd9\u4e2a$2-{\\rm by}-2$\u7684\u65cb\u8f6c\u77e9\u9635\u3002\u6211\u4eec\u5f88\u5bb9\u6613\u53d1\u73b0\u5176\u884c\u5217\u662f\u6b63\u4ea4\u7684\uff0c\u5e76\u4e14\u5176\u9006\u77e9\u9635\u7b49\u4e8e\u5176\u8f6c\u7f6e\u3002\u9006\u9635$\\rm R_{\\theta}^{-1}$\u8868\u793a\u65cb\u8f6c\u4e86$- \\theta$\u3002<\/p>\n

      \u9006\u9635$\\rm R_{\\theta}^{-1}$\u8868\u793a\u65cb\u8f6c\u4e86$- \\theta$\u3002\u6211\u4eec\u5c06\u8bc1\u660e\u8fd9\u4e00\u70b9\uff1a<\/p>\n

    \u4e09\u89d2\u51fd\u6570\u5177\u6709\uff1a<\/p>\n

    $$
    \n\\begin{eqnarray}
    \n{\\rm sin}{(- \\theta)} &=& -{\\rm sin}{(\\theta)} \\newline
    \n{\\rm cos}{(- \\theta)} &=& {\\rm cos}{(\\theta)}
    \n\\end{eqnarray}
    \n$$<\/p>\n

    \u56e0\u6b64\uff1a<\/p>\n

    $$
    \n{\\rm R_{- \\theta}} =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{{\\rm cos}{(- \\theta)}} & {-{\\rm sin}{(- \\theta)}} \\newline
    \n{{\\rm sin}{(- \\theta)}} & {{\\rm cos}{(- \\theta)}}
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{{\\rm cos}{\\theta}} & {{\\rm sin}{\\theta}} \\newline
    \n{{- \\rm sin}{\\theta}} & {{\\rm cos}{\\theta}}
    \n\\end{array}
    \n\\right) =
    \n{\\rm R_{\\theta}^{-1}}
    \n$$<\/p>\n

    \u8bc1\u6bd5\u3002<\/p>\n

    \u4e5d\u3001\u7f6e\u6362\u77e9\u9635<\/h2>\n

      \u6b63\u4ea4\u77e9\u9635\u7684\u53e6\u4e00\u4e2a\u4f8b\u5b50\u662f\u7f6e\u6362\u77e9\u9635(permutation matrices)<\/strong>\uff0c\u4e0b\u9762\u662f\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u4f8b\u5b50\uff0c\u6765\u770b\u4e00\u4e0b\u5b83\u662f\u5982\u4f55\u505a\u7f6e\u6362\u7684\u3002<\/p>\n

    \u5bf9\u4e8e\u4e00\u4e2a$2-{\\rm by}-2$\u7684\u77e9\u9635\u6765\u8bf4\uff0c\u5176\u5217\u5411\u91cf(\u6216\u8005\u884c\u5411\u91cf)\u53ef\u80fd\u7684\u6392\u5e8f\u53ea\u6709$(1,2)$\u6216\u8005$(2,1)$\uff0c\u5982\u679c\u6309\u7167\u524d\u8005\u6392\u5e8f\uff0c\u5219\u8be5\u7f6e\u6362\u77e9\u9635\u7b49\u4e8e$\\rm I$\uff0c\u5982\u679c\u6309\u540e\u8005\u6392\u5e8f\uff0c\u53ef\u4ee5\u5199\u505a\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

    \u4e0b\u9762\u662f\u5b83\u548c\u5176\u4ed6\u77e9\u9635\u8fdb\u884c\u8ba1\u7b97\u7684\u4f8b\u5b50\uff1a<\/p>\n

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

    \u5f53\u7f6e\u6362\u77e9\u9635\u4f4d\u4e8e\u5de6\u8fb9\u65f6\uff0c\u5b83\u4ea4\u6362\u4e86\u53f3\u4fa7\u77e9\u9635\u7684\u884c\u3002<\/p>\n

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

    \u5f53\u7f6e\u6362\u77e9\u9635\u4f4d\u4e8e\u53f3\u8fb9\u65f6\uff0c\u5b83\u4ea4\u6362\u4e86\u5de6\u4fa7\u77e9\u9635\u7684\u5217\u3002<\/p>\n

      \u5bf9\u4e8e\u4e00\u4e2a$3-{\\rm by}-3$\u7684\u77e9\u9635\u6765\u8bf4\uff0c\u5176\u53ef\u80fd\u5b58\u5728\u7684\u6392\u5e8f\u7ec4\u5408\u6709\uff1a$3! = 6$\u79cd\uff0c\u6211\u4eec\u89c2\u5bdf\u5176\u4e2d\u4e00\u79cd\u7ec4\u5408$(3,1,2)$\u662f\u5982\u4f55\u5f71\u54cd\u77e9\u9635\u7684\uff1a<\/p>\n

    $$
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{0} & {0} & {1} \\newline
    \n{1} & {0} & {0} \\newline
    \n{0} & {1} & {0} \\newline
    \n\\end{array}
    \n\\right)
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{a} & {b} & {c} \\newline
    \n{d} & {e} & {f} \\newline
    \n{g} & {h} & {i} \\newline
    \n\\end{array}
    \n\\right) =
    \n\\left(
    \n\\begin{array}{*{20}{c}}
    \n{g} & {h} & {i} \\newline
    \n{a} & {b} & {c} \\newline
    \n{d} & {e} & {f} \\newline
    \n\\end{array}
    \n\\right)
    \n$$<\/p>\n

    \u4e0d\u96be\u53d1\u73b0\uff0c\u7f6e\u6362\u77e9\u9635\u5c06\u53f3\u4fa7\u7684\u77e9\u9635\u7684\u884c\u4ea4\u6362\u5230\u4e86\u7f6e\u6362\u77e9\u9635\u5bf9\u5e94\u7684\u4f4d\u7f6e\u4e0a\u3002<\/p>\n

      \u503c\u5f97\u6ce8\u610f\u7684\u662f\uff0c\u7f6e\u6362\u77e9\u9635\u53ea\u662f\u901a\u8fc7\u5bf9\u5355\u4f4d\u77e9\u9635\u8fdb\u884c\u76f8\u5e94\u7684\u7f6e\u6362\u5f97\u5230\u7684\uff0c\u4e0a\u9762\u7684\u4f8b\u5b50\u5c31\u662f\u5c06$(1,2,3)$\u7f6e\u6362\u4e3a$(3,1,2)$\u5f97\u5230\u7684\u3002\u6211\u4eec\u53ef\u4ee5\u5c06\u4e00\u4e2a\u884c\u7f6e\u6362\u77e9\u9635\u8868\u793a\u4e3a\uff1a<\/p>\n

    $$
    \n\\rm PA = (PI)A
    \n$$<\/p>\n

    $\\rm P$\u662f\u7f6e\u6362\u77e9\u9635\uff0c$\\rm PI$\u662f\u5177\u6709\u7f6e\u6362\u884c\u7684\u5355\u4f4d\u77e9\u9635\u3002\u5355\u4f4d\u77e9\u9635\u662f\u6b63\u4ea4\u7684\uff0c\u56e0\u6b64\u901a\u8fc7\u5bf9\u5355\u4f4d\u77e9\u9635\u8fdb\u884c\u7f6e\u6362\u5f97\u5230\u7684\u7f6e\u6362\u77e9\u9635\u4e5f\u662f\u6b63\u4ea4\u7684\u3002\u56e0\u6b64\uff0c\u672c\u8282\u7684\u5f00\u5934\u624d\u8bf4\u7f6e\u6362\u77e9\u9635\u662f\u6b63\u4ea4\u77e9\u9635\u7684\u53e6\u4e00\u4e2a\u4f8b\u5b50\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

    \u4e00\u3001\u77e9\u9635\u5b9a\u4e49   \u4e00\u4e2a$m-{\\rm by}-n$\u7684\u77e9\u9635\u662f\u6307\u4e00\u4e2a\u5177\u6709$n$\u884c\u3001$m$\u5217\u7684\uff0c\u5305\u542b\u4e86\u6570\u5b57\u6216\u8005\u5176\u4ed6\u6570 …<\/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\/1358"}],"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=1358"}],"version-history":[{"count":1,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1358\/revisions"}],"predecessor-version":[{"id":1359,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/posts\/1358\/revisions\/1359"}],"wp:attachment":[{"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/media?parent=1358"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/categories?post=1358"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/swordofmorning.com\/index.php\/wp-json\/wp\/v2\/tags?post=1358"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}