{"id":336,"date":"2012-03-09T02:01:29","date_gmt":"2012-03-08T17:01:29","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=336"},"modified":"2021-10-03T20:51:32","modified_gmt":"2021-10-03T11:51:32","slug":"kbr200805","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr200805\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2008\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u884c\u5217 \\(A = \\left( \\begin{array}{cc} 1 & -2 \\\\ -2 & 1 \\end{array} \\right)\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(P = \\left( \\begin{array}{cc} 1 & -a \\\\ a & 1 \\end{array} \\right)\\) , \\(D = \\left( \\begin{array}{cc} x & 0 \\\\ 0 & y \\end{array} \\right)\\) \u3068\u3059\u308b. \\(AP =PD\\) \u304c\u6210\u308a\u7acb\u3064\u3068\u304d, \\(a , x , y\\) \u3092\u6c42\u3081\u3088. \u305f\u3060\u3057 \\(a \\gt 0\\) \u3068\u3059\u308b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(\\left( A +tE \\right)^n =4E\\) \u304c\u6210\u308a\u7acb\u3064\u3088\u3046\u306a\u5b9f\u6570 \\(t\\) \u3068\u81ea\u7136\u6570 \\(n\\) \u306e\u7d44\u3092\u3059\u3079\u3066\u6c42\u3081\u3088. \u305f\u3060\u3057, \\(E = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right)\\) \u3068\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

    (1)<\/strong><\/p>\r\n

    \\[\\begin{align}\r\nAP & = \\left( \\begin{array}{cc} 1 & -2 \\\\ -2 & 1 \\end{array} \\right) \\left( \\begin{array}{cc} 1 & -a \\\\ a & 1 \\end{array} \\right) = \\left( \\begin{array}{cc} -2a+1 & -a-2 \\\\ a-2 & 2a+1 \\end{array} \\right) , \\\\\r\nPD & = \\left( \\begin{array}{cc} 1 & -a \\\\ a & 1 \\end{array} \\right) \\left( \\begin{array}{cc} x & 0 \\\\ 0 & y \\end{array} \\right) = \\left( \\begin{array}{cc} x & -ay \\\\ ax & y \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6761\u4ef6\u3088\u308a\r\n\\[\r\n\\left\\{\\begin{array}{ll} x = -2a+1 & ... [1] \\\\ -ay = -a-2 & ... [2] \\\\ ax = a-2 & ... [3] \\\\ y = 2a+1 & ... [4] \\end{array}\\right.\r\n\\]\r\n[1] \u3092 [3] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n-2a^2 +a & = a-2 \\\\\r\na^2 & = 1 \\\\\r\n\\text{\u2234} \\quad a & = 1 \\quad ( \\ \\text{\u2235} \\ a \\gt 0 \\ )\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 [1] [4] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\r\nx = -1 , \\ y = 3\r\n\\]\r\n\u3053\u308c\u3089\u306f [2] \u3082\u307f\u305f\u3057\u3066\u3044\u308b.
    \r\n\u3088\u3063\u3066\r\n\\[\r\n(a,x,y) = \\underline{( 1 , -1 , 3 )}\r\n\\]\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\n\\det P = 1 \\cdot 1 -(-1) \\cdot 1 = 2\r\n\\]\r\n\u306a\u306e\u3067, \\(P\\) \u306f\u9006\u884c\u5217\u3092\u3082\u3064.
    \r\n\\(\\left( A +tE \\right)^n = 4E\\) \u306e\u4e21\u8fba\u306b\u53f3\u304b\u3089 \\(P^n\\) \u3092\u639b\u3051\u308b\u3068\r\n\\[\\begin{align}\r\n\\left( AP +tP \\right)^n & = 4P^n \\\\\r\n\\left( PD +tP \\right)^n & = 4P^n \\\\\r\nP^n \\left( D +tE \\right)^n & = 4P^n\r\n\\end{align}\\]\r\n\u4e21\u8fba\u306b\u5de6\u304b\u3089 \\(P^{-n}\\) \u3092\u639b\u3051\u308c\u3070\r\n\\[\r\n\\left( D +tE \\right)^n = 4E\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\r\n\\left( D +tE \\right)^n = \\left( \\begin{array}{cc} (t-1)^n & 0 \\\\ 0 & (t+3)^n \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\left\\{\\begin{array}{ll} (t-1)^n = 4 & ... [5] \\\\ (t+3)^n = 4 & ... [6] \\end{array}\\right.\r\n\\]\r\n\u3092\u307f\u305f\u3059 \\(t , n\\) \u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
    \r\n[5] [6] \u3088\u308a\r\n\\[\\begin{align}\r\nt = -\\sqrt[n]{4} +1 & = \\sqrt[n]{4} -3 \\\\\r\n\\text{\u2234} \\quad \\sqrt[n]{4} & = 2 \\\\\r\n\\text{\u2234} \\quad n & = 2\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \\(t = -1\\) .
    \r\n\u3088\u3063\u3066\r\n\\[\r\n(t, n) = \\underline{( -1 , 2 )}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u884c\u5217 \\(A = \\left( \\begin{array}{cc} 1 & -2 \\\\ -2 & 1 \\end{array} \\right)\\) \u306b\u3064\u3044\u3066, \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\\(P = \\left( \\beg … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[55],"tags":[144,16],"class_list":["post-336","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2008","tag-tsukuba_r","tag-16"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/336","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=336"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/336\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=336"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=336"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=336"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}