[{"@context":"http:\/\/schema.org\/","@type":"BlogPosting","@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/285639#BlogPosting","mainEntityOfPage":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/285639","headline":"\u884c\u5217\u306e\u5bfe\u6570 &#8211; Wikipedia","name":"\u884c\u5217\u306e\u5bfe\u6570 &#8211; Wikipedia","description":"before-content-x4 \u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u884c\u5217\u306e\u5bfe\u6570\uff08\u82f1\u8a9e: Logarithm of a matrix\uff09\u3068\u306f\u3001\u884c\u5217\u306e\u6307\u6570\u95a2\u6570\u3092\u65bd\u3057\u305f\u3068\u304d\u4e0e\u3048\u3089\u308c\u305f\u884c\u5217\u3068\u4e00\u81f4\u3059\u308b\u3088\u3046\u306a\u3082\u3046\u4e00\u3064\u306e\u884c\u5217\u3092\u3044\u3046\u3002\u3064\u307e\u308a\u884c\u5217\u306e\u5bfe\u6570\u51fd\u6570\u306f\u3001\u30b9\u30ab\u30e9\u30fc\u5909\u6570\u30b9\u30ab\u30e9\u30fc\u5024\u306e\u5bfe\u6570\u51fd\u6570\u306e\u4e00\u822c\u5316\u3067\u3042\u308a\u3001\u307e\u305f\u884c\u5217\u306e\u6307\u6570\u95a2\u6570\u306e\u3042\u308b\u610f\u5473\u3067\u306e\u9006\u95a2\u6570\u3092\u4e0e\u3048\u308b\u3082\u306e\u3068\u306a\u308b\u3002\u5fc5\u305a\u3057\u3082\u5168\u3066\u306e\u884c\u5217\u304c\u305d\u306e\u5bfe\u6570\u3092\u6301\u3064\u308f\u3051\u3067\u306f\u306a\u304f\u3001\u307e\u305f\u5bfe\u6570\u3092\u6301\u3064\u5834\u5408\u3067\u3042\u3063\u3066\u3082\u8907\u6570\u306e\u884c\u5217\u3092\u5bfe\u6570\u3068\u3057\u3066\u6301\u3061\u5f97\u308b\u3002\u5bfe\u6570\u3092\u6301\u3064\u884c\u5217\u306f\u4f55\u3089\u304b\u306e\u30ea\u30fc\u7fa4\u306b\u5c5e\u3057\u3001\u304b\u3064\u3001\u305d\u306e\u5bfe\u6570\u306f\u305d\u306e\u30ea\u30fc\u7fa4\u306b\u4ed8\u968f\u3059\u308b\u30ea\u30fc\u4ee3\u6570\u306e\u5143\u306b\u5bfe\u5fdc\u3059\u308b\u305f\u3081\u3001\u884c\u5217\u306e\u5bfe\u6570\u51fd\u6570\u306e\u7814\u7a76\u306f\u30ea\u30fc\u7406\u8ad6\uff08\u82f1\u8a9e\u7248\uff09\u306b\u3064\u306a\u304c\u308b\u3002 after-content-x4 \u4e0e\u3048\u3089\u308c\u305f\u6b63\u65b9\u884c\u5217 A \u306b\u5bfe\u3057\u3066\u3001eB = A \u3092\u6e80\u305f\u3059\u6b63\u65b9\u884c\u5217 B \u3092\u3001A \u306e\uff08\u884c\u5217\u306e\uff09\u5bfe\u6570\u3068\u547c\u3073 B = log(A) \u3042\u308b\u3044\u306f","datePublished":"2022-04-24","dateModified":"2022-04-24","author":{"@type":"Person","@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/author\/lordneo#Person","name":"lordneo","url":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/author\/lordneo","image":{"@type":"ImageObject","@id":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/c9645c498c9701c88b89b8537773dd7c?s=96&d=mm&r=g","height":96,"width":96}},"publisher":{"@type":"Organization","name":"Enzyklop\u00e4die","logo":{"@type":"ImageObject","@id":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","url":"https:\/\/wiki.edu.vn\/wiki4\/wp-content\/uploads\/2023\/08\/download.jpg","width":600,"height":60}},"image":{"@type":"ImageObject","@id":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/06f9de948950972540c0f6a90750bfd213c70d11","url":"https:\/\/wikimedia.org\/api\/rest_v1\/media\/math\/render\/svg\/06f9de948950972540c0f6a90750bfd213c70d11","height":"","width":""},"url":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/285639","about":["Wiki"],"wordCount":15259,"articleBody":"     (adsbygoogle = window.adsbygoogle || []).push({});before-content-x4\u6570\u5b66\u306b\u304a\u3044\u3066\u3001\u884c\u5217\u306e\u5bfe\u6570\uff08\u82f1\u8a9e: Logarithm of a matrix\uff09\u3068\u306f\u3001\u884c\u5217\u306e\u6307\u6570\u95a2\u6570\u3092\u65bd\u3057\u305f\u3068\u304d\u4e0e\u3048\u3089\u308c\u305f\u884c\u5217\u3068\u4e00\u81f4\u3059\u308b\u3088\u3046\u306a\u3082\u3046\u4e00\u3064\u306e\u884c\u5217\u3092\u3044\u3046\u3002\u3064\u307e\u308a\u884c\u5217\u306e\u5bfe\u6570\u51fd\u6570\u306f\u3001\u30b9\u30ab\u30e9\u30fc\u5909\u6570\u30b9\u30ab\u30e9\u30fc\u5024\u306e\u5bfe\u6570\u51fd\u6570\u306e\u4e00\u822c\u5316\u3067\u3042\u308a\u3001\u307e\u305f\u884c\u5217\u306e\u6307\u6570\u95a2\u6570\u306e\u3042\u308b\u610f\u5473\u3067\u306e\u9006\u95a2\u6570\u3092\u4e0e\u3048\u308b\u3082\u306e\u3068\u306a\u308b\u3002\u5fc5\u305a\u3057\u3082\u5168\u3066\u306e\u884c\u5217\u304c\u305d\u306e\u5bfe\u6570\u3092\u6301\u3064\u308f\u3051\u3067\u306f\u306a\u304f\u3001\u307e\u305f\u5bfe\u6570\u3092\u6301\u3064\u5834\u5408\u3067\u3042\u3063\u3066\u3082\u8907\u6570\u306e\u884c\u5217\u3092\u5bfe\u6570\u3068\u3057\u3066\u6301\u3061\u5f97\u308b\u3002\u5bfe\u6570\u3092\u6301\u3064\u884c\u5217\u306f\u4f55\u3089\u304b\u306e\u30ea\u30fc\u7fa4\u306b\u5c5e\u3057\u3001\u304b\u3064\u3001\u305d\u306e\u5bfe\u6570\u306f\u305d\u306e\u30ea\u30fc\u7fa4\u306b\u4ed8\u968f\u3059\u308b\u30ea\u30fc\u4ee3\u6570\u306e\u5143\u306b\u5bfe\u5fdc\u3059\u308b\u305f\u3081\u3001\u884c\u5217\u306e\u5bfe\u6570\u51fd\u6570\u306e\u7814\u7a76\u306f\u30ea\u30fc\u7406\u8ad6\uff08\u82f1\u8a9e\u7248\uff09\u306b\u3064\u306a\u304c\u308b\u3002       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4\u4e0e\u3048\u3089\u308c\u305f\u6b63\u65b9\u884c\u5217 A \u306b\u5bfe\u3057\u3066\u3001eB = A \u3092\u6e80\u305f\u3059\u6b63\u65b9\u884c\u5217 B \u3092\u3001A \u306e\uff08\u884c\u5217\u306e\uff09\u5bfe\u6570\u3068\u547c\u3073 B = log(A) \u3042\u308b\u3044\u306f ln(A) \u306a\u3069\u3067\u8868\u3059\u3002\u8907\u7d20\u6570\u306e\u5834\u5408\u3068\u540c\u69d8\u3001\u884c\u5217\u306e\u5bfe\u6570\u306f\u3057\u3070\u3057\u3070\u4e00\u610f\u3067\u306f\u306a\u3044\uff08\u5f8c\u8ff0\uff09\u3002\u306a\u304a\u3001\u6b63\u65b9\u884c\u5217\u3092\u5909\u6570\u3068\u3059\u308b\u6307\u6570\u95a2\u6570\u306f\u3001\u6b63\u65b9\u884c\u5217 B \u306b\u5bfe\u3057\u3066       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4eB=exp\u2061(B):=\u2211n=0\u221eBnn!{displaystyle e^{B}=exp(B):=sum _{n=0}^{infty }{frac {B^{n}}{n!}}}\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u6b63\u65b9\u884c\u5217 A \u306b\u5bfe\u3057\u3066\u3001B=log\u2061(c)I\u2212\u2211k=1\u221e1k(I\u22121cA)k{displaystyle B=log(c)I-sum _{k=1}^{infty }{frac {1}{k}}left(I-{frac {1}{c}}Aright)^{k}}\u304c\u9069\u5f53\u306a\u6b63\u306e\u5b9f\u6570c{displaystyle c}\u306b\u3064\u3044\u3066\u53ce\u675f\u3059\u308c\u3070\u3001       (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4B=log\u2061(A){displaystyle B=log(A)}\u3067\u3042\u308b\u3002\u8907\u7d20\u95a2\u6570log\u2061(z){displaystyle log(z)}\u306b\u3064\u3044\u3066\u3001z=c{displaystyle z=c}\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\u3001log\u2061(z)=log\u2061(c)+\u2211k=1\u221e(\u22121)k\u22121kck(z\u2212c)k=log\u2061(c)\u2212\u2211k=1\u221e1k(1\u2212zc)k{displaystyle log(z)=log(c)+sum _{k=1}^{infty }{frac {(-1)^{k-1}}{kc^{k}}}(z-c)^{k}=log(c)-sum _{k=1}^{infty }{frac {1}{k}}left(1-{frac {z}{c}}right)^{k}}\u3067\u3042\u308a\u3001\u305d\u306e\u53ce\u675f\u534a\u5f84\u306fc{displaystyle c}\u3067\u3042\u308b\u306e\u3067\u30010}&#8221;\/>\u306a\u3089\u3070c{displaystyle c}\u3092\u5341\u5206\u5927\u304d\u304f\u3068\u308c\u3070\u30c6\u30a4\u30e9\u30fc\u5c55\u958b\u306f\u53ce\u675f\u3059\u308b\u3002\u3053\u308c\u3092\u884c\u5217\u306b\u5f53\u3066\u306f\u3081\u308c\u3070\u3001\u6b63\u65b9\u884c\u5217A\u306e\u3059\u3079\u3066\u306e\u56fa\u6709\u5024(\u7279\u6027\u6839)\u306e\u5b9f\u6570\u90e8\u5206\u304c\u6b63\u3067\u3042\u308c\u3070\u3001\u9069\u5f53\u306a\u6b63\u306e\u5b9f\u6570c{displaystyle c}\u306b\u3064\u3044\u3066\u3001B=log\u2061(c)I\u2212\u2211k=1\u221e1k(I\u22121cA)k{displaystyle B=log(c)I-sum _{k=1}^{infty }{frac {1}{k}}left(I-{frac {1}{c}}Aright)^{k}}\u306f\u53ce\u675f\u3057\u3001B=log\u2061(A){displaystyle B=log(A)}\u3067\u3042\u308b\u3002Table of Contents\u4f8b: \u5e73\u9762\u56de\u8ee2\u306e\u5bfe\u6570[\u7de8\u96c6]\u3055\u3089\u306a\u308b\u4f8b: \u4e09\u6b21\u5143\u7a7a\u9593\u4e0a\u306e\u56de\u8ee2\u884c\u5217\u306e\u5bfe\u6570[\u7de8\u96c6]\u5bfe\u89d2\u5316\u53ef\u80fd\u306a\u884c\u5217\u306e\u5bfe\u6570\u306e\u8a08\u7b97\u6cd5[\u7de8\u96c6]\u5bfe\u89d2\u5316\u304c\u4e0d\u53ef\u80fd\u306a\u884c\u5217\u306e\u5bfe\u6570[\u7de8\u96c6]\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\u306e\u5bfe\u6570\u884c\u5217[\u7de8\u96c6]\u82f1\u8a9e\u7248\u3088\u308a\u306e\u76f4\u8a33[\u7de8\u96c6]\u95a2\u6570\u89e3\u6790\u5b66\u7684\u306a\u5074\u9762[\u7de8\u96c6]\u30ea\u30fc\u7fa4\u8ad6\u7684\u306a\u5074\u9762[\u7de8\u96c6]2\u00d72 \u306b\u9650\u3063\u305f\u8a71[\u7de8\u96c6]\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u51fa\u5178[\u7de8\u96c6]\u53c2\u8003\u6587\u732e[\u7de8\u96c6]\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]\u4f8b: \u5e73\u9762\u56de\u8ee2\u306e\u5bfe\u6570[\u7de8\u96c6]\u7c21\u5358\u306a\u4f8b\u304c\u3001\u5e73\u9762\u4e0a\u306e\u56de\u8ee2\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b\u3002\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3059\u308b\u89d2\u5ea6 \u03b1 \u306e\u56de\u8ee2\u306f 2\u00d72 \u884c\u5217A=[cos\u2061(\u03b1)\u2212sin\u2061(\u03b1)sin\u2061(\u03b1)cos\u2061(\u03b1)]{displaystyle A={begin{bmatrix}cos(alpha )&#038;-sin(alpha )\\sin(alpha )&#038;cos(alpha )end{bmatrix}}}\u3067\u8868\u308f\u3055\u308c\u308b\u3002\u4efb\u610f\u306e\u6574\u6570 n \u306b\u5bfe\u3057\u3066\u3001\u884c\u5217Bn=(\u03b1+2\u03c0n)[0\u2212110]{displaystyle B_{n}=(alpha +2pi n){begin{bmatrix}0&#038;-1\\1&#038;0end{bmatrix}}}\u306f A \u306e\u5bfe\u6570\u3067\u3042\u308b\u3002\u3057\u305f\u304c\u3063\u3066\u3001A \u306f\u7121\u9650\u500b\u306e\u5bfe\u6570\u3092\u6301\u3064\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u56de\u8ee2\u89d2\u304c 2\u03c0 \u306e\u6574\u6570\u500d\u306e\u9055\u3044\u3092\u9664\u3044\u3066\u3057\u304b\u6c7a\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u3068\u3044\u3046\u4e8b\u5b9f\u306b\u5bfe\u5fdc\u3059\u308b\u3082\u306e\u3067\u3042\u308b\u3002\u30ea\u30fc\u7406\u8ad6\u306e\u7528\u8a9e\u3092\u7528\u3044\u308c\u3070\u3001\u56de\u8ee2\u884c\u5217 A \u306f\u30ea\u30fc\u7fa4 SO(2) \u306e\u5143\u3067\u3042\u308a\u3001\u5bfe\u5fdc\u3059\u308b\u5bfe\u6570 B \u306f\uff08\u6b6a\u5bfe\u79f0\u884c\u5217\u5168\u4f53\u306e\u6210\u3059\uff09\u30ea\u30fc\u4ee3\u6570 \ud835\udd98\ud835\udd94(2) \u306e\u5143\u3068\u306a\u308b\u3002\u884c\u5217[01\u221210]{displaystyle {begin{bmatrix}0&#038;1\\-1&#038;0end{bmatrix}}}\u306f\u30ea\u30fc\u4ee3\u6570 \ud835\udd98\ud835\udd94(2) \u306e\u751f\u6210\u5143\u3067\u3042\u308b\u3002\u300c\u4e0e\u3048\u3089\u308c\u305f\u884c\u5217\u306b\u5bfe\u6570\u304c\u5b58\u5728\u3059\u308b\u304b\u5426\u304b\u300d\u3068\u3044\u3046\u554f\u984c\u306f\u3001\u8907\u7d20\u4fc2\u6570\u306e\u7bc4\u56f2\u3067\u8003\u3048\u308b\u3068\u304d\u306b\u6700\u3082\u5358\u7d14\u306a\u7b54\u3092\u6301\u3064\u3002\u3053\u306e\u5834\u5408\u3001\u4e0e\u3048\u3089\u308c\u305f\u884c\u5217\u304c\u5bfe\u6570\u3092\u6301\u3064\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f\u3001\u305d\u308c\u304c\u53ef\u9006\u3067\u3042\u308b\u3053\u3068\u3067\u3042\u308b[1]\u3002\u30b8\u30e7\u30eb\u30c0\u30f3\u6a19\u6e96\u5f62\u3067\u8003\u3048\u308c\u3070\u3001\u4efb\u610f\u306eA=PJP\u22121{displaystyle A=PJP^{-1}}\u306b\u5bfe\u3057\u3066\u3001exp\u2061(PXP\u22121)=\u2211n=0\u221e(PXP\u22121)nn!=P\u2211n=0\u221eXnn!P\u22121=Pexp\u2061(X)P\u22121{displaystyle exp(PXP^{-1})=sum _{n=0}^{infty }{frac {(PXP^{-1})^{n}}{n!}}=Psum _{n=0}^{infty }{frac {X^{n}}{n!}}P^{-1}=Pexp(X)P^{-1}}\u3067\u3042\u308b\u304b\u3089\u3001J=exp\u2061(X){displaystyle J=exp(X)}\u3068\u306a\u308bX{displaystyle X}\u304c\u5b58\u5728\u3059\u308c\u3070\u3001A=exp\u2061(PXP\u22121){displaystyle A=exp(PXP^{-1})}\u3068\u306a\u308aA{displaystyle A}\u306f\u5bfe\u6570\u3092\u6301\u3064\u3002\u9006\u306bA=exp\u2061(Y){displaystyle A=exp(Y)}\u3068\u306a\u308bY{displaystyle Y}\u304c\u5b58\u5728\u3059\u308c\u3070\u3001J=P\u22121AP=exp\u2061(P\u22121YP){displaystyle J=P^{-1}AP=exp(P^{-1}YP)}\u3068\u306a\u308aJ{displaystyle J}\u306f\u5bfe\u6570\u3092\u6301\u3064\u3002\u3053\u306e\u305f\u3081\u3001A{displaystyle A}\u306e\u5bfe\u6570\u306e\u5b58\u5728\u3068\u3001\u305d\u306e\u30b8\u30e7\u30eb\u30c0\u30f3\u6a19\u6e96\u5f62J{displaystyle J}\u306e\u5bfe\u6570\u306e\u5b58\u5728\u306f\u5fc5\u8981\u5341\u5206\u3067\u3042\u308b\u3002\u4e00\u65b9\u3001\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\u306b\u3064\u3044\u3066\u306f\u3001\u56fa\u6709\u5024\u304c\u30bc\u30ed\u3067\u306a\u3051\u308c\u3070\u5bfe\u6570\u884c\u5217\u3092\u6301\u3061\u3001\u56fa\u6709\u5024\u304c\u30bc\u30ed\u306a\u3089\u3070\u5bfe\u6570\u884c\u5217\u3092\u6301\u305f\u306a\u3044\u3053\u3068\u304c\u8a00\u3048\u308b\u306e\u3067\u3001\u884c\u5217A{displaystyle A}\u304c\u5bfe\u6570\u884c\u5217\u3092\u6301\u3064\u306b\u306f\u3001\u56fa\u6709\u5024\u30bc\u30ed\u3092\u6301\u305f\u306a\u3044\u3001\u5373\u3061\u884c\u5217\u5f0f\u304c\u30bc\u30ed\u3067\u306a\u3044\u3001\u5373\u3061\u53ef\u9006\u3067\u3042\u308b\u3053\u3068\u304c\u5fc5\u8981\u5341\u5206\u3068\u8a00\u3048\u308b\u3002\u5bfe\u6570\u3092\u6301\u3064\u5834\u5408\u306b\u304a\u3044\u3066\u3082\u5bfe\u6570\u304c\u4e00\u610f\u3068\u306f\u9650\u3089\u306a\u3044\u304c\u3001\u305d\u306e\u884c\u5217\u304c\u8ca0\u306e\u5b9f\u56fa\u6709\u5024\u3092\u6301\u305f\u306a\u3044\u306a\u3089\u3070\u3001\u305d\u306e\u3059\u3079\u3066\u306e\u56fa\u6709\u5024\u304c\u5e2f\u72b6\u9818\u57df {z \u2208 C | \u2212\u03c0 < \u2111m\u2009z < \u03c0} \u306b\u8f09\u3063\u3066\u3044\u308b\u3088\u3046\u306a\u5bfe\u6570\u306f\u305f\u3060\u4e00\u3064\u5b9a\u307e\u308a\u3001\u4e3b\u5024\u3042\u308b\u3044\u306f\u300c\u4e3b\u5bfe\u6570 (principal logarithm)\u300d\u3068\u547c\u3070\u308c\u308b[2]\u3002\u5b9f\u4fc2\u6570\u306e\u7bc4\u56f2\u5185\u3067\u8003\u3048\u308b\u306a\u3089\u3070\u7b54\u306f\u3088\u308a\u8fbc\u307f\u5165\u3063\u3066\u304f\u308b\u3002\u5b9f\u884c\u5217\u304c\u5b9f\u884c\u5217\u3092\u5bfe\u6570\u306b\u6301\u3064\u305f\u3081\u306e\u5fc5\u8981\u5341\u5206\u6761\u4ef6\u306f\u3001\u305d\u308c\u304c\u53ef\u9006\u304b\u3064\u8ca0\u306e\u56fa\u6709\u5024\u306b\u5c5e\u3059\u308b\u5404\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\u304c\u5076\u6570\u56de\u3042\u3089\u308f\u308c\u308b\u3053\u3068\u3067\u3042\u308b[3]\u3002\u53ef\u9006\u306a\u5b9f\u884c\u5217\u304c\u3001\u3053\u306e\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\u306b\u95a2\u3059\u308b\u6761\u4ef6\u3092\u6e80\u305f\u3055\u306a\u3044\u306a\u3089\u3070\u3001\u305d\u306e\u5bfe\u6570\u306f\u5b9f\u3067\u306a\u3044\u8907\u7d20\u884c\u5217\u306e\u4e2d\u3067\u3057\u304b\u8003\u3048\u3089\u308c\u306a\u3044\u3002\u3053\u306e\u72b6\u6cc1\u306f\u30b9\u30ab\u30e9\u30fc\u306e\u5834\u5408\u306b\u3059\u3067\u306b\u751f\u3058\u3066\u3044\u308b\u3053\u3068\u3067\u3042\u308a\u3001\u5b9f\u969b \u22121 \u306e\uff08\u8907\u7d20\uff09\u5bfe\u6570\u306f\u5b9f\u6570\u3067\u306a\u3044\u8907\u7d20\u6570\u3067\u3042\u308b\u30022\u00d72 \u5b9f\u884c\u5217\u306e\u5b9f\u5bfe\u6570\u306e\u5b58\u5728\u6027\u306b\u3064\u3044\u3066\u306f\u5f8c\u8ff0\u3059\u308b\u3002A \u304a\u3088\u3073 B \u304c\u3068\u3082\u306b\u6b63\u5b9a\u5024\u884c\u5217\u306a\u3089\u3070tr\u2061ln\u2061(AB)=tr\u2061ln\u2061(A)+tr\u2061ln\u2061(B){displaystyle operatorname {tr} {ln {(AB)}}=operatorname {tr} {ln {(A)}}+operatorname {tr} {ln {(B)}}}\u304c\u6210\u308a\u7acb\u3064\u3002A \u3068 B \u3068\u304c\u53ef\u63db\uff08AB = BA\uff09\u306a\u3068\u304dln\u2061(AB)=ln\u2061(A)+ln\u2061(B){displaystyle ln {(AB)}=ln {(A)}+ln {(B)}}\u304c\u6210\u308a\u7acb\u3064\u3002\u3053\u3053\u3067 B = A\u22121 \u3092\u4ee3\u5165\u3059\u308c\u3070ln\u2061(A\u22121)=\u2212ln\u2061(A){displaystyle ln {(A^{-1})}=-ln {(A)}}\u304c\u5f97\u3089\u308c\u308b\u3002\u3055\u3089\u306a\u308b\u4f8b: \u4e09\u6b21\u5143\u7a7a\u9593\u4e0a\u306e\u56de\u8ee2\u884c\u5217\u306e\u5bfe\u6570[\u7de8\u96c6]\u211d\u00b3 \u306b\u304a\u3051\u308b\u56de\u8ee2 R \u2208 SO(3) \u306f 3\u00d73 \u76f4\u4ea4\u884c\u5217\u306b\u3088\u3063\u3066\u4e0e\u3048\u3089\u308c\u308b\u3002\u305d\u306e\u3088\u3046\u306a\u56de\u8ee2\u884c\u5217 R \u306e\u5bfe\u6570\u306f\u30ed\u30c9\u30ea\u30b2\u30b9\u306e\u56de\u8ee2\u516c\u5f0f\u306e\u53cd\u5bfe\u79f0\u6210\u5206\u304b\u3089\u76f4\u3061\u306b\u8a08\u7b97\u3067\u304d\u308b[4] \uff08\u8ef8\u2013\u89d2\u5ea6\u8868\u73fe\uff08\u82f1\u8a9e\u7248\uff09\u3082\u53c2\u7167\uff09\u3002\u3053\u308c\u306b\u3088\u308a\u3001\u30d5\u30ed\u30d9\u30cb\u30a6\u30b9\u30ce\u30eb\u30e0\u3092\u6700\u5c0f\u3068\u3059\u308b\u5bfe\u6570\u304c\u5f97\u3089\u308c\u308b\u304c\u3001R \u304c\u56fa\u6709\u5024 \u22121 \u3092\u6301\u3064\u3068\u304d\u3001\u305d\u306e\u3088\u3046\u306a\u3082\u306e\u306f\u4e00\u610f\u3067\u306a\u3044\u305f\u3081\u3046\u307e\u304f\u3044\u304b\u306a\u3044\u3002\u3055\u3089\u306a\u308b\u6ce8\u610f\u3068\u3057\u3066\u3001\u56de\u8ee2\u884c\u5217 A, B \u306b\u5bfe\u3057\u3066\u3001dg(A,B):=\u2016log\u2061(A\u22a4B)\u2016F{displaystyle d_{g}(A,B):=|log(A^{top }B)|_{F}}\u306f\u56de\u8ee2\u884c\u5217\u5168\u4f53\u306e\u6210\u3059\u4e09\u6b21\u5143\u591a\u69d8\u4f53\u4e0a\u306e\u6e2c\u5730\u7684\u8ddd\u96e2\u3067\u3042\u308b\u3002\u5bfe\u89d2\u5316\u53ef\u80fd\u306a\u884c\u5217\u306e\u5bfe\u6570\u306e\u8a08\u7b97\u6cd5[\u7de8\u96c6]\u5bfe\u89d2\u5316\u53ef\u80fd\u884c\u5217 A \u306b\u5bfe\u3059\u308b ln\u2009A \u306e\u6c42\u3081\u65b9\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u3059\u308b\u3002\u884c\u5217 A \u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u304b\u3089\u306a\u308b\u884c\u5217 V \u3092\u6c42\u3081\u308b\uff08\u5404\u5217\u304c A \u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\uff09\u3002V \u306e\u9006\u884c\u5217 V\u22121 \u3092\u6c42\u3081\u308b\u3002\u3053\u306e\u3068\u304dA\u2032=V\u22121AV{displaystyle A&#8217;=V^{-1}AV}\u3068\u7f6e\u3051\u3070 A&#8216; \u306f A \u306e\u56fa\u6709\u5024\u304c\u5bfe\u89d2\u6210\u5206\u306b\u4e26\u3093\u3060\u5bfe\u89d2\u884c\u5217\u3068\u306a\u308b\u3002ln(A&#8216;) \u3092\u5f97\u308b\u305f\u3081\u306b\u306f\u3001A&#8216; \u306e\u5bfe\u89d2\u6210\u5206\u3092\u305d\u308c\u305e\u308c\u306e\u81ea\u7136\u5bfe\u6570\u3067\u7f6e\u304d\u63db\u3048\u308c\u3070\u3088\u3044\u3002\u3053\u308c\u306b\u3088\u308aln\u2061A=V(ln\u2061A\u2032)V\u22121{displaystyle ln A=V(ln A&#8217;)V^{-1}}\u3092\u5f97\u308b\u3002\u3053\u306e\u3088\u3046\u306a A \u306e\u5bfe\u6570\u304c\uff08A \u304c\u5b9f\u884c\u5217\u306e\u5834\u5408\u3067\u3055\u3048\uff09\u8907\u7d20\u884c\u5217\u3068\u306a\u308a\u3046\u308b\u3053\u3068\u306f\u3001\u5404\u6210\u5206\u304c\u5b9f\u304b\u3064\u6b63\u306e\u884c\u5217\u304c\uff08\u306b\u3082\u62d8\u3089\u305a\uff09\u8ca0\u306e\u3042\u308b\u3044\u306f\u3055\u3089\u306b\u8907\u7d20\u6570\u306e\u56fa\u6709\u5024\u3092\u6301\u3061\u5f97\u308b\uff08\u305d\u308c\u306f\u56de\u8ee2\u884c\u5217\u306e\u4f8b\u306b\u304a\u3044\u3066\u3082\u8a00\u3048\u308b\u3053\u3068\u3067\u3042\u308b\uff09\u3068\u3044\u3046\u4e8b\u5b9f\u304b\u3089\u5f93\u3046\u3002\u3053\u306e\u7a2e\u306e\u884c\u5217\u306e\u5bfe\u6570\u304c\u4e00\u610f\u3067\u306a\u3044\u3053\u3068\u306f\u3001\u8907\u7d20\u6570\u306e\u5bfe\u6570\u304c\u4e00\u610f\u3067\u306a\u3044\u3053\u3068\u304b\u3089\u751f\u3058\u3066\u304f\u308b\u3002\u5bfe\u89d2\u5316\u304c\u4e0d\u53ef\u80fd\u306a\u884c\u5217\u306e\u5bfe\u6570[\u7de8\u96c6]\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\u306e\u5bfe\u6570\u884c\u5217[\u7de8\u96c6]\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80deJn(\u03bb){displaystyle J_{n}(lambda )}\u3068\u306f\u3001n\u6b21\u6b63\u65b9\u884c\u5217\u3067\u3001j))ii=\u03bb{displaystyle (J_{n}(lambda ))_{ii}=lambda }\u3001(Jn(\u03bb))ii+1=1{displaystyle (J_{n}(lambda ))_{ii+1}=1}\u3001i+1}&#8221;\/>\u306e\u3068\u304d(Jn(\u03bb))ij=0{displaystyle (J_{n}(lambda ))_{ij}=0}\u3068\u306a\u308b\u884c\u5217\u3067\u3042\u308b\u3002\u03bb\u22600{displaystyle lambda neq 0}\u306e\u3068\u304d\u3001\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80deJn(\u03bb){displaystyle J_{n}(lambda )}\u306e\u5bfe\u6570\u884c\u5217log\u2061(Jn(\u03bb)){displaystyle log(J_{n}(lambda ))}\u306e\u5404\u6210\u5206\u306f\u3001j)))ij=0{displaystyle (log(J_{n}(lambda )))_{ij}=0}\u3001(log\u2061(Jn(\u03bb)))ii=log\u2061(\u03bb){displaystyle (log(J_{n}(lambda )))_{ii}=log(lambda )}\u3001i}&#8221;\/>\u306e\u3068\u304d(log\u2061(Jn(\u03bb)))ij=(\u22121)j\u2212i\u22121(j\u2212i\u22121)!\u03bb\u2212j+i{displaystyle (log(J_{n}(lambda )))_{ij}=(-1)^{j-i-1}(j-i-1)!lambda ^{-j+i}}\u3067\u3042\u308b\u3002\u3053\u306e\u3053\u3068\u306f\u3001\u6b21\u306e\u3053\u3068\u304b\u3089\u308f\u304b\u308b\u3002i}&#8221;\/>\u306e\u3068\u304d\u3001\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\u306eij{displaystyle ij}\u6210\u5206\u306f\u3001\u03bb{displaystyle lambda }\u3092\u5909\u6570\u3068\u307f\u3066\u3001ii{displaystyle ii}\u6210\u5206\u3092j\u2212i{displaystyle j-i}\u56de\u5fae\u5206\u3057\u305f\u3082\u306e\u3068\u306a\u3063\u3066\u3044\u308b\u3002\u540c\u69d8\u306e\u6027\u8cea\u306f\u3001Jn(\u03bb)k{displaystyle J_{n}(lambda )^{k}}\u3001\u5358\u4f4d\u884c\u5217\u3001\u540c\u69d8\u306e\u6027\u8cea\u3092\u6301\u3064\u884c\u5217\u306e\u5b9a\u6570\u500d\u3001\u540c\u69d8\u306e\u6027\u8cea\u3092\u6301\u3064\u884c\u5217\u3069\u3046\u3057\u306e\u548c\u306b\u3064\u3044\u3066\u3082\u6210\u308a\u7acb\u3064\u3002\u3053\u306e\u305f\u3081\u3001log\u2061(Jn(\u03bb))=log\u2061(c)I\u2212\u2211k=1\u221e1k(I\u22121cJn(\u03bb))k{displaystyle log(J_{n}(lambda ))=log(c)I-sum _{k=1}^{infty }{frac {1}{k}}left(I-{frac {1}{c}}J_{n}(lambda )right)^{k}}\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306e\u6027\u8cea\u304c\u6210\u308a\u7acb\u3064\u3002log\u2061(Jn(\u03bb)){displaystyle log(J_{n}(lambda ))}\u306e\u5bfe\u89d2\u6210\u5206\u306f\u660e\u3089\u304b\u306blog\u2061(\u03bb){displaystyle log(lambda )}\u3067\u3042\u308b\u304b\u3089\u3001\u305d\u3053\u304b\u3089\u9806\u6b21\u5fae\u5206\u3057\u3066\u4ed6\u306e\u6210\u5206\u304c\u5206\u304b\u308b\u3002\u82f1\u8a9e\u7248\u3088\u308a\u306e\u76f4\u8a33[\u7de8\u96c6]\u4e0a\u8ff0\u306e\u30a2\u30eb\u30b4\u30ea\u30ba\u30e0\u306f[1101]{displaystyle {begin{bmatrix}1&#038;1\\0&#038;1end{bmatrix}}}\u3088\u3046\u306a\u5bfe\u89d2\u5316\u4e0d\u53ef\u80fd\u306a\u884c\u5217\u306b\u3064\u3044\u3066\u306f\u9069\u7528\u3067\u304d\u306a\u3044\u3002\u3053\u306e\u3088\u3046\u306a\u884c\u5217\u306b\u5bfe\u3057\u3066\u306f\u3001\u305d\u306e\u30b8\u30e7\u30eb\u30c0\u30f3\u5206\u89e3\u3092\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u3001\u307e\u305f\u4e0a\u8ff0\u306e\u3088\u3046\u306a\u5bfe\u89d2\u6210\u5206\u306e\u5bfe\u6570\u3067\u306f\u306a\u304f\u3001\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\uff08\u82f1\u8a9e\u7248\uff09\u306e\u5bfe\u6570\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u306b\u306a\u308b\u3002\u5f8c\u8005\u306e\u4f5c\u696d\u306b\u3064\u3044\u3066\u306f\u3001\u30b8\u30e7\u30eb\u30c0\u30f3\u7d30\u80de\u304cB=[\u03bb100\u22ef00\u03bb10\u22ef000\u03bb1\u22ef0\u22ee\u22ee\u22ee\u22ee\u22f1\u22ee0000\u03bb100000\u03bb]=\u03bb[1\u03bb\u2212100\u22ef001\u03bb\u221210\u22ef0001\u03bb\u22121\u22ef0\u22ee\u22ee\u22ee\u22ee\u22f1\u22ee00001\u03bb\u22121000001]=\u03bb(I+K){displaystyle B={begin{bmatrix}lambda &#038;1&#038;0&#038;0&#038;cdots &#038;0\\0&#038;lambda &#038;1&#038;0&#038;cdots &#038;0\\0&#038;0&#038;lambda &#038;1&#038;cdots &#038;0\\vdots &#038;vdots &#038;vdots &#038;vdots &#038;ddots &#038;vdots \\0&#038;0&#038;0&#038;0&#038;lambda &#038;1\\0&#038;0&#038;0&#038;0&#038;0&#038;lambda \\end{bmatrix}}=lambda {begin{bmatrix}1&#038;lambda ^{-1}&#038;0&#038;0&#038;cdots &#038;0\\0&#038;1&#038;lambda ^{-1}&#038;0&#038;cdots &#038;0\\0&#038;0&#038;1&#038;lambda ^{-1}&#038;cdots &#038;0\\vdots &#038;vdots &#038;vdots &#038;vdots &#038;ddots &#038;vdots \\0&#038;0&#038;0&#038;0&#038;1&#038;lambda ^{-1}\\0&#038;0&#038;0&#038;0&#038;0&#038;1\\end{bmatrix}}=lambda (I+K)}\u306e\u3088\u3046\u306a\u5f62\u306b\u66f8\u304d\u8868\u305b\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3053\u3068\u3067\u9054\u6210\u3055\u308c\u308b\u3002\u3053\u3053\u3067\u3001K \u306f\u4e3b\u5bfe\u89d2\u6210\u5206\u304a\u3088\u3073\u305d\u306e\u4e0b\u304c\u3059\u3079\u3066 0 \u3067\u3042\u308b\u3088\u3046\u306a\u884c\u5217\u3067\u3042\u308b\uff08\u6570 \u03bb \u304c\u96f6\u3067\u306a\u3044\u3053\u3068\u306f\u3001\u5bfe\u6570\u304c\u53d6\u308c\u308b\u305f\u3081\u306b\u884c\u5217\u306f\u53ef\u9006\u3068\u3059\u308b\u4eee\u5b9a\u306b\u3088\u308b\uff09\u3002\u3053\u306e\u3068\u304d\u3001\u30e1\u30eb\u30ab\u30c8\u30eb\u7d1a\u6570ln\u2061(1+x)=x\u2212x22+x33\u2212x44+\u22ef{displaystyle ln(1+x)=x-{frac {x^{2}}{2}}+{frac {x^{3}}{3}}-{frac {x^{4}}{4}}+cdots }\u3092\u7528\u3044\u308c\u3070ln\u2061B=ln\u2061(\u03bb(I+K))=ln\u2061(\u03bbI)+ln\u2061(I+K)=(ln\u2061\u03bb)I+K\u2212K22+K33\u2212K44+\u22ef{displaystyle ln B=ln(lambda (I+K))=ln(lambda I)+ln(I+K)=(ln lambda )I+K-{frac {K^{2}}{2}}+{frac {K^{3}}{3}}-{frac {K^{4}}{4}}+cdots }\u3092\u5f97\u308b\u3002\u4e00\u822c\u306b\u306f\u3001\u3053\u306e\u7d1a\u6570\u306f\u4efb\u610f\u306e\u884c\u5217 K \u306b\u5bfe\u3057\u3066\u53ce\u675f\u3059\u308b\u308f\u3051\u3067\u306f\u306a\u3044\uff08\u7d76\u5bfe\u5024\u304c 1 \u3088\u308a\u5927\u304d\u3044\u4efb\u610f\u306e\u5b9f\u6570\u306b\u5bfe\u3057\u3066\u53ce\u675f\u3057\u306a\u3044\u306e\u3068\u540c\u69d8\uff09\u304c\u3001\u4eca\u306e\u5834\u5408\u306b\u9650\u3063\u3066\u306f K \u306f\u51aa\u96f6\u884c\u5217\u3067\u3042\u308b\u304b\u3089\u3001\u5b9f\u969b\u306b\u306f\u6709\u9650\u9805\u3057\u304b\u306a\u3044\uff08K \u306e\u6b21\u5143\u304c m \u306a\u3089 Km \u306f\u96f6\u884c\u5217\u3067\u3042\u308b\uff09\u3002\u3053\u306e\u3084\u308a\u65b9\u3067\u3001\u4f8b\u3048\u3070ln[1101]=[0100]{displaystyle ln !{begin{bmatrix}1&#038;1\\0&#038;1end{bmatrix}}={begin{bmatrix}0&#038;1\\0&#038;0end{bmatrix}}}\u3092\u5f97\u308b\u3002\u95a2\u6570\u89e3\u6790\u5b66\u7684\u306a\u5074\u9762[\u7de8\u96c6]\u6b63\u65b9\u884c\u5217\u306f\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593 Rn\uff08n \u306f\u884c\u5217\u306e\u6b21\u5143\uff09\u4e0a\u306e\u7dda\u5f62\u4f5c\u7528\u7d20\u3092\u8868\u73fe\u3059\u308b\u3002\u305d\u306e\u3088\u3046\u306a\u7a7a\u9593\u306f\u6709\u9650\u6b21\u5143\u3067\u3042\u308b\u304b\u3089\u3001\u3053\u306e\u4f5c\u7528\u7d20\u306f\u5b9f\u969b\u306b\u6709\u754c\u3067\u3042\u308b\u3002\u6b63\u5247\u6c4e\u51fd\u6570\u8a08\u7b97\uff08\u82f1\u8a9e\u7248\uff09\u306e\u9053\u5177\u7acb\u3066\u3092\u7528\u3044\u308b\u3068\u3001\u8907\u7d20\u6570\u5e73\u9762\u5185\u306e\u958b\u96c6\u5408\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u305f\u6b63\u5247\u95a2\u6570 f(z) \u304a\u3088\u3073\u6709\u754c\u4f5c\u7528\u7d20 T \u306b\u5bfe\u3057\u3001f(z) \u304c T \u306e\u30b9\u30da\u30af\u30c8\u30eb\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u9650\u308a\u306b\u304a\u3044\u3066\u3001f(T) \u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u95a2\u6570 f(z) = ln\u2009z \u306f\u3001\u8907\u7d20\u6570\u5e73\u9762\u5185\u306e\u539f\u70b9\u3092\u542b\u307e\u306a\u3044\u4efb\u610f\u306e\u5358\u9023\u7d50\u958b\u96c6\u5408\u4e0a\u3067\u5b9a\u7fa9\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u304b\u3064\u305d\u306e\u3088\u3046\u306a\u9818\u57df\u4e0a\u3067\u6b63\u5247\u3067\u3042\u308b\u3002\u3053\u306e\u3053\u3068\u306f T \u306e\u30b9\u30da\u30af\u30c8\u30eb\u304c\u539f\u70b9\u3092\u542b\u307e\u305a\u3001\u539f\u70b9\u304b\u3089\u7121\u9650\u9060\u70b9\u3078\u5411\u304b\u3046 T \u306e\u30b9\u30da\u30af\u30c8\u30eb\u3092\u6a2a\u5207\u3089\u306a\u3044\u5f84\u8def\u304c\u5b58\u5728\u3059\u308b\u306a\u3089\u3070 ln\u2009T \u304c\u5b9a\u7fa9\u3067\u304d\u308b\u3053\u3068\u3092\u793a\u3057\u3066\u3044\u308b\uff08\u4f8b\u3048\u3070\u3001T \u306e\u30b9\u30da\u30af\u30c8\u30eb\u304c\u305d\u306e\u5185\u5074\u306b\u539f\u70b9\u304c\u304f\u308b\u3088\u3046\u306a\u5186\u306a\u3089\u3070\u3001ln\u2009T \u306f\u5b9a\u7fa9\u3067\u304d\u306a\u3044\uff09\u3002\u30e6\u30fc\u30af\u30ea\u30c3\u30c9\u7a7a\u9593\u306e\u5834\u5408\u306b\u7acb\u3061\u623b\u308b\u3068\u3001\u3053\u306e\u7a7a\u9593\u4e0a\u306e\u7dda\u5f62\u4f5c\u7528\u7d20\u306e\u30b9\u30da\u30af\u30c8\u30eb\u306f\u305d\u306e\u8868\u73fe\u884c\u5217\u306e\u56fa\u6709\u5024\u5168\u4f53\u306e\u6210\u3059\u96c6\u5408\u3067\u3042\u308a\u3001\u305d\u308c\u306f\u6709\u9650\u96c6\u5408\u3067\u3042\u308b\u3002\u305d\u306e\u30b9\u30da\u30af\u30c8\u30eb\u306b\u539f\u70b9\u304c\u542b\u307e\u308c\u306a\u3044\uff08\u884c\u5217\u304c\u53ef\u9006\uff09\u3067\u3042\u308b\u9650\u308a\u306b\u304a\u3044\u3066\u3001\u524d\u6bb5\u843d\u3067\u8ff0\u3079\u305f\u5f84\u8def\u306b\u95a2\u3059\u308b\u6761\u4ef6\u306a\u3069\u306f\u660e\u3089\u304b\u306b\u6e80\u305f\u3055\u308c\u308b\u306e\u3067\u3001\u305d\u306e\u8ad6\u6cd5\u306b\u3088\u308a ln\u2009T \u304c\u5b9a\u7fa9\u53ef\u80fd\u3067\u3042\u308b\u3002\u3053\u306e\u7a2e\u306e\u884c\u5217\u306e\u5bfe\u6570\u304c\u4e00\u610f\u3067\u306a\u3044\u3053\u3068\u306f\u3001\u884c\u5217\u306e\u56fa\u6709\u5024\u96c6\u5408\u4e0a\u3067\u5b9a\u7fa9\u3055\u308c\u308b\u5bfe\u6570\u51fd\u6570\u306e\u5206\u679d\u304c\u8907\u6570\u9078\u3073\u3046\u308b\u3068\u3044\u3046\u4e8b\u5b9f\u304b\u3089\u751f\u3058\u308b\u3002\u30ea\u30fc\u7fa4\u8ad6\u7684\u306a\u5074\u9762[\u7de8\u96c6]\u30ea\u30fc\u7fa4\u8ad6\u306b\u304a\u3044\u3066\u3001\u30ea\u30fc\u4ee3\u6570 \ud835\udd24 \u304b\u3089\u5bfe\u5fdc\u3059\u308b\u30ea\u30fc\u7fa4 G \u3078\u306e\u6307\u6570\u5199\u50cfexp:g\u2192G{displaystyle exp colon {mathfrak {g}}to G}\u304c\u5b58\u5728\u3059\u308b\u3002\u884c\u5217\u30ea\u30fc\u7fa4\u306b\u5bfe\u3057\u3066\u3001\ud835\udd24 \u304a\u3088\u3073 G \u306e\u5143\u306f\u6b63\u65b9\u884c\u5217\u3067\u3042\u308a\u3001\u6307\u6570\u5199\u50cf\u306f\u884c\u5217\u306e\u6307\u6570\u95a2\u6570\u3067\u4e0e\u3048\u3089\u308c\u308b\u3002\u305d\u306e\u9006\u5199\u50cf log\u00a0:= exp\u22121 \u306f\u591a\u4fa1\u3067\u3042\u308a\u3001\u672c\u9805\u3067\u6271\u3046\u884c\u5217\u306e\u5bfe\u6570\u3068\u4e00\u81f4\u3059\u308b\u3002\u5bfe\u6570\u5199\u50cf\u306f\u30ea\u30fc\u7fa4 G \u3092\u4ed8\u968f\u3059\u308b\u30ea\u30fc\u4ee3\u6570 \ud835\udd24 \u3078\u5199\u3059\u3002\u3053\u3053\u3067\u3001\u6307\u6570\u5199\u50cf\u306f\u96f6\u884c\u5217 0 \u2208 \ud835\udd24 \u306e\u8fd1\u508d U \u3068\u5358\u4f4d\u884c\u5217 1 \u2208 G \u306e\u8fd1\u508d V \u306e\u9593\u306e\u5c40\u6240\u5fae\u5206\u540c\u76f8\u5199\u50cf\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b[5]\u3002\u3057\u305f\u304c\u3063\u3066\u3001\uff08\u884c\u5217\u306e\uff09\u5bfe\u6570\u51fd\u6570\u306flog:V(\u2282G)\u2192U(\u2282g){displaystyle log colon V_{(subset G)}to U_{(subset {mathfrak {g}})}}\u306a\u308b\u5199\u50cf\u3068\u3057\u3066\u77db\u76fe\u306a\u304f\u5b9a\u7fa9\u3055\u308c\u308b\u3002\u3053\u306e\u3068\u304d\u3001\u30e4\u30b3\u30d3\u306e\u516c\u5f0f\uff08\u82f1\u8a9e\u7248\uff09\u306e\u91cd\u8981\u306a\u7cfb\u3068\u3057\u3066log\u2061(det(A))=tr\u2061(log\u2061A){displaystyle log(det(A))=operatorname {tr} (log A)}\u304c\u6210\u308a\u7acb\u3064\u30022\u00d72 \u306b\u9650\u3063\u305f\u8a71[\u7de8\u96c6]2\u00d72 \u5b9f\u884c\u5217\u304c\u8ca0\u306e\u884c\u5217\u5f0f\u3092\u6301\u3064\u3068\u304d\u3001\u305d\u306e\u5b9f\u5bfe\u6570\u306f\u5b58\u5728\u3057\u306a\u3044\u3002\u307e\u305a\u521d\u3081\u306b\u3001\u4efb\u610f\u306e 2\u00d72 \u5b9f\u884c\u5217\u306f\u4e09\u7a2e\u985e\u306e\u8907\u7d20\u6570 z = x + y\u03b5\uff08\u305f\u3060\u3057 \u03b5\u00b2 \u2208 {\u22121, 0, +1}\uff09\u306e\u3044\u305a\u308c\u304b\u4e00\u7a2e\u985e\u3068\u898b\u306a\u3059\u3053\u3068\u304c\u3067\u304d\u3066\u3001\u305d\u306e\u3068\u304d\u306e z \u306f2\u00d72 \u5b9f\u884c\u5217\u5168\u4f53\u306e\u6210\u3059\u74b0\u306e\u90e8\u5206\u8907\u7d20\u6570\u5e73\u9762\u4e0a\u306e\u70b9\u306b\u306a\u3063\u3066\u3044\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308b\u3002\u884c\u5217\u5f0f\u304c\u8ca0\u3067\u3042\u308b\u3088\u3046\u306a\u5834\u5408\u306f \u03b5\u00b2 = +1 \u306e\u5834\u5408\u3001\u3059\u306a\u308f\u3061\u5206\u89e3\u578b\u8907\u7d20\u6570\u5e73\u9762\u4e0a\u306b\u3057\u304b\u5b58\u5728\u3057\u306a\u3044\u3002\u3053\u306e\u5e73\u9762\u306e\u3046\u3061\u306e1\/4\u306e\u307f\u304c\u6307\u6570\u5199\u50cf\u306e\u50cf\u3067\u3042\u3063\u3066\u3001\u3053\u306e\u90e8\u5206\uff08\u56db\u5206\u8c61\u9650\uff09\u306b\u304a\u3044\u3066\u306e\u307f\u5bfe\u6570\u5199\u50cf\u304c\u5b9a\u7fa9\u3067\u304d\u308b\u3002\u4e09\u3064\u3042\u308b\u4ed6\u306e\u8c61\u9650\u306f \u03b5 \u3068 \u22121 \u304c\u751f\u6210\u3059\u308b\u30af\u30e9\u30a4\u30f3\u306e\u56db\u5143\u7fa4\u306e\u4f5c\u7528\u306b\u3088\u308b\u4e00\u3064\u76ee\u306e\u8c61\u9650\u306e\u50cf\u306b\u306a\u308b\u3002\u305f\u3068\u3048\u3070\u3001a = ln\u20092\uff08\u3053\u306e\u3068\u304d cosh\u2009a = 5\/4 \u304a\u3088\u3073 sinh\u2009a = 3\/4\uff09\u3068\u3059\u308c\u3070\u3001\u884c\u5217\u306e\u5f62\u3067A=exp[0aa0]=[cosh\u2061asinh\u2061asinh\u2061acosh\u2061a]=[1.25.75.751.25]{displaystyle A=exp !{begin{bmatrix}0&#038;a\\a&#038;0end{bmatrix}}={begin{bmatrix}cosh a&#038;sinh a\\sinh a&#038;cosh aend{bmatrix}}={begin{bmatrix}1.25&#038;.75\\.75&#038;1.25end{bmatrix}}}\u3068\u66f8\u304f\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3089\u3001\u3053\u306e\u884c\u5217\u306fln\u2061A=[0ln\u20612ln\u206120]{displaystyle ln A={begin{bmatrix}0&#038;ln 2\\ln 2&#038;0end{bmatrix}}}\u3092\u5bfe\u6570\u306b\u6301\u3064\u3002\u3057\u304b\u3057\u3001\u4ee5\u4e0b\u306e\u884c\u5217[3\/45\/45\/43\/4],\u00a0[\u22123\/4\u22125\/4\u22125\/4\u22123\/4],\u00a0[\u22125\/4\u22123\/4\u22123\/4\u22125\/4]{displaystyle {begin{bmatrix}3\/4&#038;5\/4\\5\/4&#038;3\/4end{bmatrix}}, {begin{bmatrix}-3\/4&#038;-5\/4\\-5\/4&#038;-3\/4end{bmatrix}}, {begin{bmatrix}-5\/4&#038;-3\/4\\-3\/4&#038;-5\/4end{bmatrix}}}.\u306f\u5bfe\u6570\u3092\u6301\u305f\u306a\u3044\u3002\u3053\u308c\u3089\u306f\u3001\u4e0a\u8ff0\u306e\u56db\u5143\u7fa4\u306e\u4f5c\u7528\u306e\u4e0b\u3067\u5bfe\u6570\u3092\u6301\u3064\u4e0a\u8a18\u306e\u884c\u5217 A \u306e\u5171\u8edb\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u307b\u304b\u306e\u4e09\u3064\u3092\u8868\u3057\u3066\u3044\u308b\u3002\u6b63\u5247\u306a 2\u00d72 \u5b9f\u884c\u5217 2 x 2 \u884c\u5217\u304c\u5fc5\u305a\u3057\u3082\u5bfe\u6570\u3092\u6301\u3064\u3068\u306f\u9650\u3089\u306a\u3044\u304c\u3001\u3053\u306e\u56db\u5143\u7fa4\u306b\u3088\u308b\u4f5c\u7528\u306e\u3082\u3068\u5bfe\u6570\u3092\u6301\u3064\u884c\u5217\u306b\u5171\u5f79\u306b\u306a\u308b\u3002\u307e\u305f\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u3053\u3068\u3082\u5f93\u3046\u3002\u305f\u3068\u3048\u3070\u3001\u4e0a\u8ff0\u306e\u884c\u5217 A \u306e\u5e73\u65b9\u6839\uff08\u82f1\u8a9e\u7248\uff09\u306f\u6307\u6570\u51fd\u6570\u306b (ln\u2009A)\/2 \u3092\u4ee3\u5165\u3059\u308b\u3053\u3068\u306b\u3088\u308a\u3001\u76f4\u63a5\u7684\u306bA=[cosh\u2061((ln\u20612)\/2)sinh\u2061((ln\u20612)\/2)sinh\u2061((ln\u20612)\/2)cosh\u2061((ln\u20612)\/2)]=[1.06.35.351.06]{displaystyle {sqrt {A}}={begin{bmatrix}cosh((ln 2)\/2)&#038;sinh((ln 2)\/2)\\sinh((ln 2)\/2)&#038;cosh((ln 2)\/2)end{bmatrix}}={begin{bmatrix}1.06&#038;.35\\.35&#038;1.06end{bmatrix}}}\u3068\u8a08\u7b97\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u3002\u3088\u308a\u8c4a\u304b\u306a\u4f8b\u3068\u3057\u3066\u3001\u521d\u3081\u306b \u30d4\u30bf\u30b4\u30e9\u30b9\u306e\u4e09\u3064\u7d44\uff08\u82f1\u8a9e\u7248\uff09 (p, q, r) \u3092\u3068\u3063\u3066 a = ln(p + r) \u2212 ln\u2009q \u3068\u304a\u304f\u3068ea=p+rq=cosh\u2061a+sinh\u2061a{displaystyle e^{a}={frac {p+r}{q}}=cosh a+sinh a}\u304c\u6210\u308a\u7acb\u3064\u3002\u3059\u308b\u3068\u3044\u307eexp[0aa0]=[r\/qp\/qp\/qr\/q]{displaystyle exp !{begin{bmatrix}0&#038;a\\a&#038;0end{bmatrix}}={begin{bmatrix}r\/q&#038;p\/q\\p\/q&#038;r\/qend{bmatrix}}}\u3068\u306a\u308b\u304b\u30891q[rppr]{displaystyle {frac {1}{q}}{begin{bmatrix}r&#038;p\\p&#038;rend{bmatrix}}}\u306f\u884c\u5217[0aa0](a=ln\u2061(p+r)\u2212ln\u2061q){displaystyle {begin{bmatrix}0&#038;a\\a&#038;0end{bmatrix}}quad (a=ln(p+r)-ln q)}\u3092\u5bfe\u6570\u306b\u6301\u3064\u3002\u95a2\u9023\u9805\u76ee[\u7de8\u96c6]\u51fa\u5178[\u7de8\u96c6]^ Higham (2008), Theorem 1.27^ Higham (2008), Theorem 1.31^ Culver (1966)^ Eng\u00f8 (2001)^ Hall 2015 Theorem 3.42\u53c2\u8003\u6587\u732e[\u7de8\u96c6]Gantmacher, Felix R. (1959), The Theory of Matrices, 1, New York: Chelsea, pp.\u00a0239\u2013241\u00a0.Hall, Brian C. (2015), Lie Groups, Lie Algebras, and Representations An Elementary Introduction, Graduate Texts in Mathematics, 222 (2nd ed.), Springer, ISBN\u00a00-387-40122-9\u00a0Culver, Walter J. (1966), \u201cOn the existence and uniqueness of the real logarithm of a matrix\u201d, Proceedings of the American Mathematical Society 17 (5):  1146\u20131151, doi:10.1090\/S0002-9939-1966-0202740-6, ISSN\u00a00002-9939\u00a0.Higham, Nicholas (2008), Functions of Matrices. Theory and Computation, SIAM, ISBN\u00a0978-0-89871-646-7\u00a0.Eng\u00f8, Kenth (June 2001), \u201cOn the BCH-formula in so(3)\u201d, BIT Numerical Mathematics 41 (3):  629\u2013632, doi:10.1023\/A:1021979515229, ISSN\u00a00006-3835, http:\/\/www.ii.uib.no\/publikasjoner\/texrap\/abstract\/2000-201.html\u00a0\u5916\u90e8\u30ea\u30f3\u30af[\u7de8\u96c6]     (adsbygoogle = window.adsbygoogle || []).push({});after-content-x4"},{"@context":"http:\/\/schema.org\/","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/#breadcrumbitem","name":"Enzyklop\u00e4die"}},{"@type":"ListItem","position":2,"item":{"@id":"https:\/\/wiki.edu.vn\/jp\/wiki9\/archives\/285639#breadcrumbitem","name":"\u884c\u5217\u306e\u5bfe\u6570 &#8211; Wikipedia"}}]}]