[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/328291#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki10\/archives\/328291","headline":"\u591a\u91cd\u7dda\u578b\u4ee3\u6570 &#8211; Wikipedia","name":"\u591a\u91cd\u7dda\u578b\u4ee3\u6570 &#8211; Wikipedia","description":"before-content-x4 \u6570\u5b66\u306b\u304a\u3051\u308b\u591a\u91cd\u7dda\u578b\u4ee3\u6570\uff08\u305f\u3058\u3085\u3046\u305b\u3093\u3051\u3044\u3060\u3044\u3059\u3046\u3001\u82f1\u8a9e: multilinear algebra\uff09\u3068\u306f\u3001\u7dda\u578b\u7a7a\u9593\u306b\u304a\u3051\u308b\u591a\u91cd\u7dda\u578b\u6027 (multilinearity) \u3092\u6271\u3046\u4ee3\u6570\u5b66\u306e\u5206\u91ce\u3002\u591a\u91cd\u7dda\u578b\u6027\u306f\u5178\u578b\u7684\u306b\u306f\u7dda\u578b\u74b0\u306b\u304a\u3051\u308b\u7a4d\u306e\u69cb\u9020\u306b\u73fe\u308c\u3066\u3044\u308b\u3002A \u3092 K \u2013\u4ee3\u6570\u3068\u3059\u308b\u3068\u304d\u3001\u81ea\u7136\u6570 n \u306b\u5bfe\u3057\u3001A \u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f n \u5909\u6570\u5199\u50cf (x 1, &#8230;, xn) \u2192 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\u3092\u53ef\u63db\u74b0\u3068\u3059\u308b\u3002Table of Contents\u7279\u5fb4\u4ed8\u3051[\u7de8\u96c6]\u30c6\u30f3\u30bd\u30eb\u4ee3\u6570[\u7de8\u96c6]\u5bfe\u79f0\u4ee3\u6570[\u7de8\u96c6]\u5916\u7a4d\u4ee3\u6570[\u7de8\u96c6]\u69cb\u6210[\u7de8\u96c6]\u30c6\u30f3\u30bd\u30eb\u7a4d\u3068\u30c6\u30f3\u30bd\u30eb\u4ee3\u6570[\u7de8\u96c6]\u5bfe\u79f0\u4ee3\u6570\u3068\u5bfe\u79f0\u7a4d[\u7de8\u96c6]\u5916\u7a4d\u4ee3\u6570\u3068\u5916\u7a4d[\u7de8\u96c6]\u570f\u3068\u95a2\u624b\u306b\u3088\u308b\u8a00\u3044\u63db\u3048[\u7de8\u96c6]\u5bfe\u79f0\u4ee3\u6570\u3084\u5916\u7a4d\u4ee3\u6570\u306e\u69cb\u9020[\u7de8\u96c6]\u591a\u9805\u5f0f\u74b0[\u7de8\u96c6]\u5e7e\u4f55\u5b66\u3078\u306e\u5fdc\u7528[\u7de8\u96c6]\u7269\u7406\u5b66\u3078\u306e\u5fdc\u7528[\u7de8\u96c6]\u30d5\u30a9\u30c3\u30af\u7a7a\u9593[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u7279\u5fb4\u4ed8\u3051[\u7de8\u96c6]\u30c6\u30f3\u30bd\u30eb\u4ee3\u6570[\u7de8\u96c6]K \u2013\u52a0\u7fa4 E \u306e\u30c6\u30f3\u30bd\u30eb\u4ee3\u6570 TE \u3068\u306f\u3001\u53ef\u63db\u3068\u306f\u9650\u3089\u306a\u3044 K \u2013\u4ee3\u6570\u3067\u3042\u3063\u3066 E \u304b\u3089\u306e\u7dda\u578b\u5199\u50cf E \u2192 TE \u3092\u6301\u3061\u3001\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3082\u306e\u306e\u3053\u3068\u3067\u3042\u308b\uff1a\uff08\u53ef\u63db\u3068\u306f\u9650\u3089\u306a\u3044\uff09K \u2013\u4ee3\u6570 A \u3078\u306e K \u2013\u7dda\u578b\u5199\u50cf E \u2192 A \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\u56f3\u5f0fE\u2192A\u2193\u2193TE\u2192A{displaystyle {begin{array}{ccc}E&#038;to &#038;A\\downarrow &#038;&#038;downarrow \\mathrm {T} E&#038;to &#038;Aend{array}}}       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u304c\u53ef\u63db\u306b\u306a\u308b\u3088\u3046\u306a K \u2013\u4ee3\u6570\u306e\u6e96\u540c\u578b TE \u2192 A \u304c\u5b58\u5728\u3057\u3066\u4e00\u610f\u306b\u5b9a\u307e\u308b\u3002\u3053\u306e\u6761\u4ef6\u306b\u3088\u3063\u3066\u5bfe (TE , E \u2192 TE ) \u306f\u540c\u578b\u3092\u9664\u304d\u4e00\u610f\u306b\u5b9a\u307e\u308b\u3002\u5bfe\u79f0\u4ee3\u6570[\u7de8\u96c6]K \u2013\u52a0\u7fa4 E \u306e\u5bfe\u79f0\u4ee3\u6570 SE \u3068\u306f\u3001\u53ef\u63db\u306a K \u2013\u4ee3\u6570\u3067\u3042\u3063\u3066 E \u304b\u3089\u306e K \u2013\u7dda\u578b\u5199\u50cf\u3092\u3082\u3061\u3001\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3082\u306e\u306e\u3053\u3068\u3067\u3042\u308b\uff1a\u53ef\u63db K \u2013\u4ee3\u6570 A \u3078\u306e K \u2013\u7dda\u578b\u5199\u50cf E \u2192 A \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001\u56f3\u5f0f       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4E\u2192A\u2193\u2193SE\u2192A{displaystyle {begin{array}{ccc}E&#038;to &#038;A\\downarrow &#038;&#038;downarrow \\mathrm {S} E&#038;to &#038;Aend{array}}}\u304c\u53ef\u63db\u306b\u306a\u308b\u3088\u3046\u306a K \u2013\u4ee3\u6570\u306e\u6e96\u540c\u578b SE \u2192 A \u304c\u5b58\u5728\u3057\u3066\u4e00\u610f\u306b\u5b9a\u307e\u308b\u3002\u3053\u306e\u6761\u4ef6\u306b\u3088\u3063\u3066\u5bfe (SE , E \u2192 TE ) \u306f\u540c\u578b\u3092\u9664\u304d\u4e00\u610f\u306b\u5b9a\u307e\u308b\u3002\u5916\u7a4d\u4ee3\u6570[\u7de8\u96c6]K \u2013\u52a0\u7fa4 E \u306e\u5916\u7a4d\u4ee3\u6570 \u2227E \u3068\u306f\u3001\u53ef\u63db\u3068\u306f\u9650\u3089\u306a\u3044 K \u2013\u4ee3\u6570\u3067\u3042\u3063\u3066 E \u304b\u3089\u306e K \u2013\u7dda\u578b\u5199\u50cf\u3092\u6301\u3061\u3001\u6b21\u306e\u6761\u4ef6\u3092\u6e80\u305f\u3059\u3082\u306e\u306e\u3053\u3068\u3067\u3042\u308b\uff1a\uff08\u53ef\u63db\u3068\u306f\u9650\u3089\u306a\u3044\uff09K \u2013\u4ee3\u6570\u3078\u306e\u7dda\u578b\u5199\u50cf\u3067\u3001\u4efb\u610f\u306e x \u2208 E \u306b\u3064\u3044\u3066\u3001\u03d5(x)2\u00a0=\u00a00{displaystyle {phi (x)}^{2}~=~0}\u3068\u306a\u3063\u3066\u3044\u308b\u3082\u306e\u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u306b\u3001\u56f3\u5f0f       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4E\u2192A\u2193\u2193\u22c0E\u2192A{displaystyle {begin{array}{ccc}E&#038;to &#038;A\\downarrow &#038;&#038;downarrow \\{boldsymbol {bigwedge }}E&#038;to &#038;Aend{array}}}\u304c\u53ef\u63db\u306b\u306a\u308b\u3088\u3046\u306a K \u2013\u4ee3\u6570\u306e\u6e96\u540c\u578b\u304c\u5b58\u5728\u3057\u3066\u4e00\u610f\u306b\u5b9a\u307e\u308b\u3002\u3053\u306e\u6761\u4ef6\u306b\u3088\u3063\u3066\u5bfe (\u2227E , E \u2192 TE ) \u306f\u540c\u578b\u3092\u9664\u304d\u4e00\u610f\u306b\u5b9a\u307e\u308b\u3002\u69cb\u6210[\u7de8\u96c6]\u30c6\u30f3\u30bd\u30eb\u7a4d\u3068\u30c6\u30f3\u30bd\u30eb\u4ee3\u6570[\u7de8\u96c6]T0E = K \u3068\u3057\u30011 < n \u306b\u3064\u3044\u3066 n \u56de\u30c6\u30f3\u30bd\u30eb\u7a4d\u3092\u3068\u3063\u305f\u3082\u306e\u3092TnE = E \u2297n = E \u2297 \u2026 \u2297 E \u3068\u3057\u3001\u3053\u308c\u3089\u306e\u76f4\u548c \u2295TnE \u3092 TE \u3068\u3059\u308b\u3002\u3053\u306e K \u2013\u52a0\u7fa4\u306fTmE\u00d7TnE\u2192Tm+nE,(x1\u2297\u22ef\u2297xm,y1\u2297\u22ef\u2297yn)\u2192x1\u2297\u22ef\u2297xm\u2297y1\u2297\u22ef\u2297yn{displaystyle {mathrm {T} ^{m}E}times {mathrm {T} ^{n}E}to mathrm {T} ^{m+n}E,,left(x_{1}otimes dots otimes x_{m},,y_{1}otimes dots otimes y_{n}right)to x_{1}otimes dots otimes x_{m}otimes y_{1}otimes dots otimes y_{n}}\u306b\u3088\u3063\u3066\u5b9a\u307e\u308b\u7a4d\u3092\u6301\u3061\uff08\u4e00\u822c\u306b\u306f\u975e\u53ef\u63db\u306a\uff09K \u2013\u4ee3\u6570\u306b\u306a\u308b\u3002TnE \u3092 E \u306e n \u6b21\u30c6\u30f3\u30bd\u30eb\u51aa (n th tensor power) \u3068\u547c\u3076\u3002E \u304b\u3089 TE \u3078\u306e\u7dda\u578b\u5199\u50cf\u306f E = T1E \u2192 TE \u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u3066\u3044\u308b\u3002E \u304b\u3089 K \u2013\u4ee3\u6570 A \u3078\u306e K \u2013\u7dda\u578b\u5199\u50cf \u03c6: E \u2192 A \u304c\u4e0e\u3048\u3089\u308c\u305f\u3068\u304d\u3001E \u2192 TE \u3068\u4e21\u7acb\u3059\u308b\u6e96\u540c\u578b TE \u2192 A \u306f x 1 \u2297 \u2026 \u2297 xm \u2192 \u03c6x 1 \u2297 \u2026 \u2297 \u03c6xm \u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b\u3002\u5bfe\u79f0\u4ee3\u6570\u3068\u5bfe\u79f0\u7a4d[\u7de8\u96c6]\u30c6\u30f3\u30bd\u30eb\u4ee3\u6570 TE \u306b\u304a\u3044\u3066 x \u2297 y \u2212 y \u2297 x (x, y \u2208 E ) \u3068\u3044\u3046\u5f62\u306e T2E \u306e\u5143\u304c\u751f\u6210\u3059\u308b\u4e21\u5074\u30a4\u30c7\u30a2\u30eb\u3092 IE \u3068\u3059\u308b\u3002\u5546\u74b0 SE = TE\/IE \u3068 K \u2013\u6e96\u540c\u578b E \u2192 TE \u2192 SE \u306f\u4e0a\u306b\u6319\u3052\u305f\u5bfe\u79f0\u4ee3\u6570\u306e\u666e\u904d\u6027\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3002SE \u306b\u304a\u3051\u308b TnE \u306e \u50cf SnE \u3092 E \u306e n \u6b21\u5bfe\u79f0\u51aa (n th symmetric product) \u3068\u547c\u3076\u3002\u76f4\u63a5\u7684\u306b\u306f\u3001SnE \u306f TnE \u3092\u305d\u306e\u90e8\u5206\u52a0\u7fa4\u27e8a(x\u2297y\u2212y\u2297x)b|{displaystyle leftlangle a(xotimes y-yotimes x)bright|} a,b{displaystyle a,b} \u306f\u6589\u6b21\u5143\u3067 deg\u2061(a)+deg\u2061(b)=n\u22122\u27e9{displaystyle left.deg(a)+deg(b)=n-2rightrangle }\u3067\u5272\u3063\u305f\u5546\u52a0\u7fa4\u3068\u306a\u3063\u3066\u304a\u308a\u3001SE \u306f SnE \u306e\u76f4\u548c\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u5916\u7a4d\u4ee3\u6570\u3068\u5916\u7a4d[\u7de8\u96c6]x \u2297 x \u3068\u3044\u3046\u5f62\u306e\u5143\u304c\u751f\u6210\u3059\u308b\u4e21\u5074\u30a4\u30c7\u30a2\u30eb\u3092 JE \u3068\u3059\u308b\u3002\u5546\u74b0 \u2227E = TE\/JE \u3068 K \u2013\u6e96\u540c\u578b E \u2192 TE \u2192 \u2227E \u306f\u4e0a\u306b\u6319\u3052\u305f\u5bfe\u79f0\u4ee3\u6570\u306e\u666e\u904d\u6027\u3092\u6e80\u305f\u3057\u3066\u3044\u308b\u3002TnE \u306e\u50cf \u2227nE \u3092 E \u306e n \u6b21\u5916\u51aa (n th exterior product) \u3068\u547c\u3076\u3002\u76f4\u63a5\u7684\u306b\u306f \u2227nE \u306f TnE \u3092\u305d\u306e\u90e8\u5206\u52a0\u7fa4\u27e8a(x\u2297x)b|{displaystyle leftlangle a(xotimes x)bright|} a,b{displaystyle a,b} \u306f\u6589\u6b21\u5143\u3067 deg\u2061(a)+deg\u2061(b)=n\u22122\u27e9{displaystyle left.deg(a)+deg(b)=n-2rightrangle }\u3067\u5272\u3063\u305f\u5546\u52a0\u7fa4\u3068\u306a\u3063\u3066\u304a\u308a\u3001\u2227E \u306f \u2227nE \u306e\u76f4\u548c\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u570f\u3068\u95a2\u624b\u306b\u3088\u308b\u8a00\u3044\u63db\u3048[\u7de8\u96c6]\u4e0a\u306b\u6319\u3052\u305f\u30c6\u30f3\u30bd\u30eb\u4ee3\u6570\u306e\u7279\u5fb4\u4ed8\u3051\u306f\u3001E \u2192 T(E ) \u304c K \u2013\u4ee3\u6570\u306e\u570f\u304b\u3089 K \u2013\u52a0\u7fa4\u306e\u570f\u3078\u306e\u57cb\u3081\u8fbc\u307f\u95a2\u624b\u306e\u5de6\u968f\u4f34\u95a2\u624b\u3067\u3042\u308b\u3053\u3068\u3092\u3044\u3063\u3066\u3044\u308b\u3002\u540c\u69d8\u306b\u3057\u3066 E \u2192 S (E ) \u306f\u53ef\u63db K \u2013\u4ee3\u6570\u306e\u570f\u304b\u3089 K \u52a0\u7fa4\u306e\u570f\u3078\u306e\u57cb\u3081\u8fbc\u307f\u95a2\u624b\u306e\u5de6\u968f\u4f34\u95a2\u624b\u306b\u306a\u3063\u3066\u3044\u308b\u3002\u30c6\u30f3\u30bd\u30eb\u7a4d\u52a0\u7fa4\u3084\u5bfe\u79f0\u7a4d\u52a0\u7fa4\u3001\u5916\u7a4d\u52a0\u7fa4\u306b\u3064\u3044\u3066\u3082\u95a2\u624b\u7684\u306a\u7279\u5fb4\u4ed8\u3051\u304c\u3067\u304d\u308b\u3002n \u6b21\u30c6\u30f3\u30bd\u30eb\u51aa\u306f n \u5909\u6570\u53cc\u7dda\u578b\u5199\u50cf\u3092\u8868\u73fe\u3057\u3066\u3044\u308b\u3002\u3064\u307e\u308a\u3001K \u2013\u52a0\u7fa4 F \u306b\u5bfe\u3057\u3066 E \u304b\u3089 F \u3078\u306e n \u91cd\u7dda\u578b\u5199\u50cf\u3092 Ln (E ; F ) \u3068\u66f8\u304f\u3053\u3068\u306b\u3059\u308c\u3070\u3001\u95a2\u624b\u306e\u9593\u306e\u81ea\u7136\u306a\u540c\u4e00\u8996 Ln (E ; F ) = HomK (TnE , F ) \u304c\u3042\u308b\u3002\u540c\u69d8\u306b\u3057\u3066 n \u6b21\u5bfe\u79f0\u51aa\u3084 n \u6b21\u5916\u51aa\u3082\u305d\u308c\u305e\u308c\u3042\u308b\u95a2\u624b\u3092\u8868\u73fe\u3057\u3066\u3044\u308b\u3068\u898b\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u5177\u4f53\u7684\u306b\u306f\u3001SnE \u306f n \u6b21\u5bfe\u79f0\u5199\u50cf\u306e\u7a7a\u9593Symn\u2061(E;F)={\u03d5{displaystyle operatorname {Sym} _{n}(E,;F)={phi } \u306f E{displaystyle E} \u304b\u3089 F{displaystyle F} \u3078\u306e n{displaystyle n} \u91cd\u7dda\u578b\u5199\u50cf\u3067 \u03d5(x1,\u2026,xi,xi+1,\u2026,xn)=\u03d5(x1,\u2026,xi+1,xi,\u2026,xn){displaystyle phi left(x_{1},dots ,x_{i},x_{i+1},dots ,x_{n}right)=phi left(x_{1},dots ,x_{i+1},x_{i},dots ,x_{n}right)} \u3092\u6e80\u305f\u3059\u3002}{displaystyle }}\u3092 Symn (E ; F ) \u2261 HomK (SnE , F ) \u3068\u3057\u3066\u8868\u73fe\u3057\u3066\u3044\u308b\u3002\u540c\u69d8\u306b\u3057\u3066 \u2227nE \u306f n \u6b21\u4ea4\u4ee3\u5199\u50cf\u306e\u7a7a\u9593Altn\u2061(E;F)={\u03d5{displaystyle operatorname {Alt} _{n}(E,;F)={phi } \u306f E{displaystyle E} \u304b\u3089 F{displaystyle F} \u3078\u306e n{displaystyle n} \u91cd\u7dda\u578b\u5199\u50cf\u3067 xi=xi+1{displaystyle x_{i}=x_{i+1}} \u306a\u3089\u3070 \u03d5(x1,\u2026,xn)=0{displaystyle phi left(x_{1},dots ,x_{n}right)=0} \u3092\u6e80\u305f\u3059\u3002}{displaystyle }}\u3092\u8868\u73fe\u3057\u3066\u3044\u308b\u3002\u5bfe\u79f0\u4ee3\u6570\u3084\u5916\u7a4d\u4ee3\u6570\u306e\u69cb\u9020[\u7de8\u96c6]\u52a0\u7fa4\u306e\u76f4\u548c E \u2295 F \u306b\u5bfe\u3057\u3066\u3001\u6b21\u6570\u4ed8\u304d\u52a0\u7fa4\u3068\u3057\u3066\u306e\u81ea\u7136\u306a\u540c\u4e00\u8996 S (E \u2295 F )  \u2261 SE \u2297 SF \u3084 \u2227(E \u2295 F )  \u2261 \u2227E \u2297 \u2227F \u304c\u3042\u308b\u3002\u3064\u307e\u308a\u3001\u5404\u81ea\u7136\u6570 k \u306b\u3064\u3044\u3066Sk(E\u2295F)\u2261\u2a01k=m+nSmE\u2297SnF,\u00a0\u22c0k(E\u2295F)\u2261\u2a01k=m+n\u22c0mE\u2297\u22c0nF{displaystyle mathrm {S} ^{k}(Eoplus F)equiv {boldsymbol {bigoplus }}_{k=m+n}mathrm {S} ^{m}Eotimes mathrm {S} ^{n}F,~{boldsymbol {bigwedge }}^{k}(Eoplus F)equiv {boldsymbol {bigoplus }}_{k=m+n}{boldsymbol {bigwedge }}^{m}Eotimes {boldsymbol {bigwedge }}^{n}F}\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001dim SnE \u3084 dim \u2227nE \u306e\u6bcd\u95a2\u6570 \u03c3t (E ) = \u2211 dim(SnE )tn \u3084 \u03bbt (E ) = \u2211 dim(\u2227nE )tn \u306b\u3064\u3044\u3066\u03c3t(E\u2295F)=\u03c3t(E)\u03c3t(F),\u00a0\u03bbt(E\u2295F)=\u03bbt(E)\u03bbt(F){displaystyle sigma _{t}(Eoplus F)=sigma _{t}(E)sigma _{t}(F),~lambda _{t}(Eoplus F)=lambda _{t}(E)lambda _{t}(F)}\u304c\u6210\u7acb\u3057\u3066\u3044\u308b\u3002\u3053\u3053\u304b\u3089 \u03c3t (K ) = 1 + t + t 2 + \u2026 = 1\/(1 &#8211; t ) \u3084 \u03bbt (K ) = 1 + t \u304b\u3089 dim \u2227nKm = nCm \u306a\u3069\u304c\u5f93\u3046\u3002\u591a\u9805\u5f0f\u74b0[\u7de8\u96c6]n \u6b21\u306e\u81ea\u7531 K \u2013\u52a0\u7fa4\uff08K \u304c\u4f53\u306e\u3068\u304d\u306b\u306f n 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