{"id":4739,"date":"2024-08-13T05:41:03","date_gmt":"2024-08-13T05:41:03","guid":{"rendered":"https:\/\/science-hub.click\/%E7%89%B9%E6%80%A7%E5%A4%9A%E9%A0%85%E5%BC%8F%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC\/"},"modified":"2024-08-13T05:41:03","modified_gmt":"2024-08-13T05:41:03","slug":"%E7%89%B9%E6%80%A7%E5%A4%9A%E9%A0%85%E5%BC%8F%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=4739","title":{"rendered":"\u7279\u6027\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac"},"content":{"rendered":"

\u7dda\u5f62\u4ee3\u6570\u3067\u306f\u3001\u6b63\u65b9\u884c\u5217\u307e\u305f\u306f\u6709\u9650\u6b21\u5143\u30d9\u30af\u30c8\u30eb\u7a7a\u9593\u306e\u6e96\u540c\u578b\u6027\u306f\u3001\u7279\u6027\u591a\u9805\u5f0f<\/strong>\u3068\u547c\u3070\u308c\u308b\u591a\u9805\u5f0f\u306b\u95a2\u9023\u4ed8\u3051\u3089\u308c\u307e\u3059\u3002\u3053\u308c\u306b\u306f\u3001\u56fa\u6709\u5024\u3001\u884c\u5217\u5f0f\u3001\u30c8\u30ec\u30fc\u30b9\u306a\u3069\u3001\u884c\u5217\u307e\u305f\u306f\u6e96\u540c\u578b\u6027\u306b\u95a2\u3059\u308b\u91cd\u8981\u306a\u60c5\u5831\u304c\u542b\u307e\u308c\u3066\u3044\u307e\u3059\u3002<\/p>

\u30e2\u30c1\u30d9\u30fc\u30b7\u30e7\u30f3<\/span><\/span><\/h2>

\u6b21\u6570n<\/i>\u306e\u6b63\u65b9\u884c\u5217M<\/i>\u304c\u4e0e\u3048\u3089\u308c\u305f\u5834\u5408\u3001\u6839\u304c\u6b63\u78ba\u306bM<\/i>\u306e\u56fa\u6709\u5024\u3067\u3042\u308b\u591a\u9805\u5f0f<\/a><\/span>\u3092\u898b\u3064\u3051\u305f\u3044\u3068\u3057\u307e\u3059\u3002<\/p>

M<\/i>\u304c\u5bfe\u89d2\u884c\u5217<\/a><\/span>\u3001\u3088\u308a\u4e00\u822c\u7684\u306b\u306f\u4e09\u89d2\u884c\u5217<\/a><\/span>\u306e\u5834\u5408\u3001 M<\/i>\u306e\u56fa\u6709\u5024\u3001\u03bb 1<\/sub> \u3001…\u3001\u03bb n \u306f<\/i><\/sub>M<\/i>\u306e\u5bfe\u89d2\u4fc2\u6570\u3067\u3042\u308a\u3001 \u7279\u6027\u591a\u9805\u5f0f\u3092<\/a><\/span>\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {(X-\\lambda_1)(X-\\lambda_2)\\ldots(X-\\lambda_n)\\quad} $$<\/div><\/dd><\/dl>

\u3053\u306e\u591a\u9805\u5f0f\u306f\u884c\u5217\u5f0fdet( X<\/i> I<\/i> n<\/i><\/sub> \u2212 M<\/i> )<\/span>\u3067\u3042\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3053\u3053\u3067\u3001 I<\/i> n<\/i><\/sub>\u306f\u5358\u4f4d\u884c\u5217\u3067\u3059\u3002<\/p>

\u4efb\u610f\u306e\u884c\u5217M<\/i>\u306b\u3064\u3044\u3066\u3001 \u03bb \u304cM<\/i>\u306e\u56fa\u6709\u5024<\/span>\u3067\u3042\u308b\u5834\u5408\u3001 MV<\/i> = \u03bb V<\/i> \u3001\u3064\u307e\u308a (\u03bb I<\/i> n<\/i><\/sub> \u2212 M<\/i> ) V<\/i> = 0 \u3068\u306a\u308b\u975e\u30bc\u30ed\u306e\u56fa\u6709\u5217V<\/i>\u304c\u5b58\u5728\u3059\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059 (\u3053\u3053\u3067\u3001 I<\/i> n<\/i><\/sub>\u306fV \u304c<\/i>\u30bc\u30ed\u3067\u306f\u306a\u3044\u305f\u3081\u3001\u3053\u308c\u306f\u884c\u5217 \u03bb I<\/i> n<\/i><\/sub> – M<\/i>\u304c\u7279\u7570\u3067\u3042\u308a\u3001\u3057\u305f\u304c\u3063\u3066\u884c\u5217\u5f0f\u304c 0 \u3067\u3042\u308b\u3053\u3068\u3092\u610f\u5473\u3057\u307e\u3059\u3002 M<\/i>\u306e\u56fa\u6709\u5024\u304c\u95a2\u6570\u03bb<\/i>\u306e\u30bc\u30ed\u3067\u3042\u308b\u3053\u3068\u3092\u8a3c\u660e\u3057\u305f\u3068\u3053\u308d\u3067\u3059\u3002 det( \u03bb\u30fbI n<\/sub> \u2212 M<\/i> )<\/span>\u307e\u305f\u306f\u591a\u9805\u5f0fdet( X<\/i> I<\/i> n<\/i><\/sub> \u2212 M<\/i> )<\/span>\u306e\u6839\u3002<\/p>

\n\"\u7279\u6027\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u6b63\u5f0f\u306a\u5b9a\u7fa9<\/a><\/span><\/span><\/h2>

M \u3092<\/i>\u53ef\u63db\u30ea\u30f3\u30b0\u5185\u306e\u4fc2\u6570\u3092\u6301\u3064\u6b21\u6570n<\/i>\u306e\u6b63\u65b9\u884c\u5217\u3068\u3057\u307e\u3059\u3002 p<\/i> M<\/i><\/sub> ( X<\/i> ) \u3067\u793a\u3055\u308c\u308bM<\/i>\u306e\u7279\u6027\u591a\u9805\u5f0f\u306f<\/strong>\u3001\u6b21\u306e\u3088\u3046\u306b\u5b9a\u7fa9\u3055\u308c\u308b\u591a\u9805\u5f0f\u3067\u3059\u3002<\/p>

<\/dd><\/dl>

\u3053\u3053\u3067\u3001 I<\/i> n \u306f<\/i><\/sub>\u6b21\u6570n<\/i>\u306e\u5358\u4f4d\u884c\u5217\u3092\u793a\u3057\u307e\u3059\u3002\u884c\u5217\u306e\u884c\u5217\u5f0f\u306f\u7a4d\u306e\u548c\u3068\u3057\u3066\u5b9a\u7fa9\u3055\u308c\u308b\u305f\u3081\u3001 p<\/i> M \u306f<\/i><\/sub>\u78ba\u304b\u306b\u591a\u9805\u5f0f\u3067\u3059\u3002<\/p>

\u6c17\u3065\u3044\u305f<\/dt>
\u7279\u6027\u591a\u9805\u5f0f\u3092det( M<\/i> \u2212 X<\/i> I<\/i> n<\/i><\/sub> )<\/span>\u3068\u3057\u3066\u5b9a\u7fa9\u3059\u308b\u8457\u8005\u3082\u3044\u307e\u3059\u304c\u30012 \u3064\u306e\u591a\u9805\u5f0f\u306f\u7b26\u53f7\u304c 1 \u3064\u3060\u3051\u7570\u306a\u308b\u305f\u3081\u3001\u3053\u308c\u306f\u91cd\u8981\u3067\u306f\u3042\u308a\u307e\u305b\u3093\u3002\u7279\u6027\u591a\u9805\u5f0f\u304c\u30e6\u30cb\u30bf\u30ea\u3068\u306a\u308b\u3088\u3046\u306b\u3001\u4ed6\u306e\u5b9a\u7fa9\u3092\u9078\u629e\u3057\u307e\u3057\u305f\u3002<\/i><\/dd><\/dl>

\u4f8b<\/span><\/h2>

\u884c\u5217\u306e\u7279\u6027\u591a\u9805\u5f0f\u3092\u6c7a\u5b9a\u3057\u305f\u3044\u3068\u3057\u307e\u3059\u3002 <\/p>

$$ {M=\\begin{pmatrix} 2 & 1\\\\ -1& 0 \\end{pmatrix}} $$<\/div><\/dd><\/dl>

\u884c\u5217\u306e\u884c\u5217\u5f0f\u3092\u8a08\u7b97\u3059\u308b\u5fc5\u8981\u304c\u3042\u308a\u307e\u3059<\/p>

$$ {XI_2-M = \\begin{pmatrix} X-2&-1\\\\ 1&X \\end{pmatrix}} $$<\/div><\/dd><\/dl>

\u305d\u3057\u3066\u3053\u308c\u306f\u6b21\u3068\u7b49\u3057\u3044<\/p>

$$ {(X-2)(X)-1(-1) = X^2-2X+1\\quad} $$<\/div><\/dd><\/dl>

\u3053\u306e\u6700\u5f8c\u306e\u591a\u9805\u5f0f\u306f\u3001 M<\/i>\u306e\u7279\u6027\u591a\u9805\u5f0f\u3067\u3059\u3002<\/p>

\u6b21\u306e\u5f0f\u3082\u4f7f\u7528\u3067\u304d\u307e\u3059<\/p>

$$ {X^2 – \\operatorname{tr}(M)X + \\det(M)\\quad} $$<\/div><\/dd><\/dl>

\u6b21\u5143<\/a><\/span>(2,2) \u306e\u884c\u5217\u306e\u5834\u5408\u3002<\/p>

\n\"\u7279\u6027\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u30d7\u30ed\u30d1\u30c6\u30a3<\/span><\/h2>

\u591a\u9805\u5f0fp<\/i> M<\/i><\/sub> ( t<\/i> ) \u306f\u30e6\u30cb\u30bf\u30ea (\u652f\u914d\u7684\u306a\u4fc2\u6570<\/a><\/span>\u306f 1 \u306b\u7b49\u3057\u3044) \u3067\u3042\u308a\u3001\u305d\u306e\u6b21\u6570<\/span>\u306fn<\/i>\u306b\u7b49\u3057\u3044\u3002\u7279\u6027\u591a\u9805\u5f0f\u306e\u6700\u3082<\/i>\u91cd\u8981\u306a\u7279\u6027\u306f\u3001 M<\/i>\u306e<\/i>\u56fa\u6709\u5024<\/i>\u304c\u307e\u3055\u306b<\/i><\/sub>\u591a\u9805\u5f0f<\/i><\/sub>p<\/i> -1) \u306e\u6839\u3067\u3042\u308a\u3001 M<\/i>\u306e\u884c\u5217\u5f0f\u3092n<\/i><\/sup>\u500d\u3057\u305f\u3082\u306e\u3067\u3042\u308a\u3001 X<\/i> n<\/i> -1<\/sup>\u306e\u4fc2\u6570\u304c\u6b21\u306e\u30c8\u30ec\u30fc\u30b9\u306e\u9006<\/span>\u306b\u7b49\u3057\u3044\u3068\u3044\u3046\u3053\u3068\u3067\u3059\u3002 M.<\/i><\/p>

\u6b21\u6570 2 \u306e\u884c\u5217M<\/i>\u306e\u5834\u5408\u3001\u7279\u6027\u591a\u9805\u5f0f\u306f\u5358\u7d14\u306b\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002 <\/p>

$$ {X^2 – \\operatorname{tr}(M)X+ \\det(M)} $$<\/div><\/dd><\/dl>

\u3053\u3053\u3067\u3001 tr( M<\/i> ) \u306fM<\/i>\u306e\u884c\u5217\u306e\u30c8\u30ec\u30fc\u30b9\u3092\u8868\u3057\u3001 det( M<\/i> ) \u306fM<\/i>\u306e\u884c\u5217\u5f0f\u3092\u8868\u3057\u307e\u3059\u3002<\/p>

\u6b21\u6570 3 \u306e\u884c\u5217A<\/i>\u306e\u5834\u5408\u3001\u7279\u6027\u591a\u9805\u5f0f\u306f\u5358\u7d14\u306b\u6b21\u306e\u3088\u3046\u306b\u8868\u3055\u308c\u307e\u3059\u3002 <\/p>

$$ {X^3 – \\operatorname{tr}(A)X^2 + Z(A)X – \\det(A)} $$<\/div><\/dd><\/dl>

\u307e\u305f\u306f<\/p>

$$ {Z(A)= -\\frac{1}{2}\\Bigl(\\operatorname{tr}(A^2) – \\bigl(\\operatorname{tr}(A)\\bigr)^2\\Bigr) = (a_{1,1} a_{2,2} + a_{1,1} a_{3,3} + a_{2,2} a_{3,3} ) – ( a_{2,1} a_{1,2} + a_{3,1} a_{1,3} + a_{3,2} a_{2,3})} $$<\/div> \u3001<\/dd><\/dl>

a<\/i> i<\/i> , j<\/i>\u306f<\/sub><\/span>\u884c\u5217A<\/i>\u306e\u4f4d\u7f6e (i, j) \u306e\u8981\u7d20\u3067\u3059\u3002<\/p>

\u4e00\u822c\u306b\u3001\u6b21\u6570 n \u3067\u306f\u3001\u6700\u5927\u6b21\u6570\u306e\u9805\u306e\u307f\u304c\u5bfe\u8c61\u3068\u306a\u308a\u3001\u6b21\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002 <\/p>

$$ {X^n – \\operatorname{tr}(A) \\times X^{n-1} + … + (-1)^n \\times \\det(A)} $$<\/div><\/dd><\/dl>

\u30b1\u30a4\u30ea\u30fc<\/a><\/span>–<\/i>\u30cf\u30df\u30eb\u30c8\u30f3<\/i><\/sub>\u306e<\/i>\u5b9a\u7406<\/i>\u306f<\/i><\/strong><\/i>\u3001<\/i>\u200b<\/i>\u200b<\/i><\/sub>\u3057\u305f\u304c\u3063\u3066\u3001\u6700\u5c0f\u591a\u9805\u5f0f\u304c<\/a><\/span>M<\/i>\u306e\u7279\u6027\u591a\u9805\u5f0f\u3092\u5206\u5272\u3059\u308b\u3053\u3068\u3092\u8a3c\u660e\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>

2 \u3064\u306e\u985e\u4f3c\u3057\u305f\u884c\u5217\u306f\u3001\u540c\u3058\u7279\u6027\u591a\u9805\u5f0f\u3092\u6301\u3061\u307e\u3059\u3002\u9006\u306f<\/a><\/span>\u4e00\u822c\u306b\u5f53\u3066\u306f\u307e\u308a\u307e\u305b\u3093\u3002\u540c\u3058\u7279\u6027\u591a\u9805\u5f0f\u3092\u6301\u3064 2 \u3064\u306e\u884c\u5217\u306f\u3001\u5fc5\u305a\u3057\u3082\u985e\u4f3c\u3057\u3066\u3044\u308b\u3068\u306f\u9650\u308a\u307e\u305b\u3093\u3002<\/p>

\u884c\u5217M<\/i>\u3068\u305d\u306e\u8ee2\u7f6e\u306f\u540c\u3058\u7279\u6027\u591a\u9805\u5f0f\u3092\u6301\u3061\u307e\u3059\u3002<\/p>

\u884c\u5217M \u306f<\/i>\u3001\u305d\u306e\u7279\u6027\u591a\u9805\u5f0f\u304c\u6b21\u6570 1 \u306e\u4fc2\u6570\u3068\u6b21\u6570\u306e\u4fc2\u6570\u306e\u7a4d\u306b\u5b8c\u5168\u306b<\/span>\u5206\u89e3\u3067\u304d\u308b\u5834\u5408\u306b\u9650\u308a\u3001\u4e09\u89d2\u884c\u5217\u306b\u4f3c\u3066\u3044\u307e\u3059\u3002

$$ {\\mathbb{K}} $$<\/div> \u3002<\/p>

\u5b9f\u969b\u3001\u3053\u306e\u5834\u5408\u3001 M \u306f<\/i>\u30b8\u30e7\u30eb\u30c0\u30f3\u884c\u5217\u306b\u3055\u3048\u4f3c\u3066\u3044\u307e\u3059\u3002<\/p>

\u30b3\u30f3\u30d1\u30cb\u30aa\u30f3\u30de\u30c8\u30ea\u30c3\u30af\u30b9<\/span><\/span><\/h2>

\u3069\u3061\u3089\u304b

$$ {p(X)=X^n-\\sum_{k=1}^n a_{n-k}X^{n-k}} $$<\/div>\u4fc2\u6570\u3092\u542b\u3080\u591a\u9805\u5f0f
$$ {\\mathbb{K}} $$<\/div> \u3002 n<\/i>\u6b21\u306e\u884c\u5217<\/p>
$$ {M=\\begin{pmatrix} 0 & 1 & 0 & \\ldots & \\ldots & 0 \\\\ \\vdots & \\ddots & \\ddots & \\ddots & & \\vdots \\\\ \\vdots & & \\ddots & \\ddots & \\ddots & \\vdots \\\\ \\vdots & & & \\ddots & \\ddots & 0 \\\\ 0 & \\ldots & \\ldots & \\ldots & 0 & 1 \\\\ a_0 & a_1 & a_2 & \\ldots &a_{n-2} & a_{n-1} \\end{pmatrix}} $$<\/div><\/dd><\/dl>

p(X) \u3092<\/i>\u7279\u6027\u591a\u9805\u5f0f (\u304a\u3088\u3073\u6700\u5c0f\u591a\u9805\u5f0f) \u3068\u3057\u3066\u8a8d\u3081\u308b\u3082\u306e\u306f\u3001\u591a\u9805\u5f0f\u306e\u968f\u4f34\u884c\u5217<\/strong>(\u307e\u305f\u306f\u7279\u5b9a\u306e\u8457\u4f5c\u306b\u3088\u308c\u3070\u3001\u305d\u306e\u8ee2\u7f6e\u884c\u5217) \u3068\u547c\u3070\u308c\u307e\u3059\u3002\u6570\u5024\u8a08\u7b97<\/a><\/span>\u3067\u591a\u9805\u5f0f\u306e\u6839\u306e\u8fd1\u4f3c\u5024\u3092\u8a08\u7b97\u3059\u308b\u65b9\u6cd5\u306e 1 \u3064\u306f\u3001\u30b3\u30f3\u30d1\u30cb\u30aa\u30f3\u884c\u5217\u3092\u69cb\u7bc9\u3057\u3001\u305d\u306e\u5f8c\u53cd\u5fa9\u6cd5\u3092\u4f7f\u7528\u3057\u3066\u3053\u306e\u884c\u5217\u306e\u56fa\u6709\u5024\u306e\u8fd1\u4f3c\u5024\u3092\u8a08\u7b97\u3059\u308b\u3053\u3068\u3067\u3059\u3002<\/p><\/div>

\n\"\u7279\u6027\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\"<\/figure>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. \u0645\u062a\u0639\u062f\u062f\u0629 \u062d\u062f\u0648\u062f \u0645\u0645\u064a\u0632\u0629 \u2013 arabe<\/a><\/li>
  2. Polinomi caracter\u00edstic \u2013 catalan<\/a><\/li>
  3. \u041c\u0430\u0442\u0440\u0438\u0446\u0103\u043d \u043a\u0103\u0442\u0430\u0440\u0442\u0430\u0432\u043b\u0430 \u043f\u043e\u043b\u0438\u043d\u043e\u043c\u0115 \u2013 tchouvache<\/a><\/li>
  4. Charakteristisches Polynom \u2013 allemand<\/a><\/li>
  5. Characteristic polynomial \u2013 anglais<\/a><\/li>
  6. Karakteriza ekvacio \u2013 esp\u00e9ranto<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u7279\u6027\u591a\u9805\u5f0f\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n