{"id":881,"date":"2013-08-19T00:02:13","date_gmt":"2013-08-18T15:02:13","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=881"},"modified":"2021-10-20T14:20:27","modified_gmt":"2021-10-20T05:20:27","slug":"ykr201302","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ykr201302\/","title":{"rendered":"\u6a2a\u56fd\u5927\u7406\u7cfb2013\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306f \\(A^2 = A\\) \u3092\u6e80\u305f\u3059.\r\n\u884c\u5217 \\(B\\) \u306f \\(B \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} a \\\\ 1 \\end{array} \\right)\\) , \\(B^2 \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ 0 \\end{array} \\right)\\) \u3092\u6e80\u305f\u3059. \u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(a+d\\) , \\(ad-bc\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(B\\) \u3092 \\(a\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(c = 1\\) \u306e\u3068\u304d, \u5b9f\u6570 \\(s , t\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n( sA+tB )^n = x _ n A +y _ n B \\quad ( n = 1 , 2, 3 , \\cdots )\r\n\\]\r\n\u3068\u8868\u3055\u308c\u308b\u3053\u3068\u3092\u793a\u3057, \\(x _ n , y _ n\\) \u3092 \\(s , t , n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \u6761\u4ef6\u3092, \u30b1\u30fc\u30ea\u30fc\u30fb\u30cf\u30df\u30eb\u30c8\u30f3\u306e\u5b9a\u7406\u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\nA -(a+d)A +(ad-bc)E = O \\\\\r\n\\text{\u2234} \\quad (1-a-d)A = (ad-bc)E \\quad ... [1]\r\n\\end{gather}\\]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(A = kE\\) \uff08 \\(k\\) \u306f\u5b9f\u6570\uff09... [2] \u3068\u8868\u305b\u308b\u3068\u304d
      \r\n[2] \u3092\u6761\u4ef6\u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\nk^2 E & = k E \\\\\r\n\\text{\u2234} \\quad k(k-1) E & = O \\\\\r\n\\text{\u2234} \\quad k & = 0 , 1\r\n\\end{align}\\]\r\n\\(k = 0\\) \u306e\u3068\u304d, \\(A = O\\) \u306a\u306e\u3067\r\n\\[\r\n( a+d , ad-bc ) = ( 0 , 0 )\r\n\\]\r\n\\(k = 1\\) \u306e\u3068\u304d, \\(A = E\\) \u306a\u306e\u3067\r\n\\[\r\n( a+d , ad-bc ) = ( 2 , 1 )\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(A \\neq kE\\) \uff08 \\(k\\) \u306f\u5b9f\u6570\uff09\u306e\u3068\u304d
      \r\n[1] \u3088\u308a\r\n\\[\\begin{align}\r\n1-a-d = 0 , & \\ ad-bc = 0 \\\\\r\n\\text{\u2234} \\quad ( a+d , ad-bc ) & = ( 1 , 0 )\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a\r\n\\[\r\n( a+d , ad-bc ) = \\underline{( 0 , 0 ) , ( 2 , 1 ) , ( 1 , 0 )}\r\n\\]\r\n

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

      \u6761\u4ef6\u3088\u308a\r\n\\[\r\nB^2 \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = B \\left( \\begin{array}{c} a \\\\ 1 \\end{array} \\right) = \\left( \\begin{array}{c} 0 \\\\ 0 \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nB \\left( \\begin{array}{cc} 1 & a \\\\ 0 & 1 \\end{array} \\right) = \\left( \\begin{array}{cc} a & 0 \\\\ 1 & 0 \\end{array} \\right) \\quad ... [3]\r\n\\]\r\n\u3053\u3053\u3067, \u9006\u884c\u5217\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{cc} 1 & a \\\\ 0 & 1 \\end{array} \\right)^{-1} & = \\dfrac{1}{1 \\cdot 1 -0 \\cdot a} \\left( \\begin{array}{cc} 1 & -a \\\\ 0 & 1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 1 & -a \\\\ 0 & 1 \\end{array} \\right)\r\n\\end{align}\\]\r\n\u304c\u5b58\u5728\u3059\u308b\u306e\u3067, [3] \u306e\u4e21\u8fba\u53f3\u304b\u3089\u639b\u3051\u3066\r\n\\[\\begin{align}\r\nB & = \\left( \\begin{array}{cc} a & 0 \\\\ 1 & 0 \\end{array} \\right) \\left( \\begin{array}{cc} 1 & -a \\\\ 0 & 1 \\end{array} \\right) \\\\\r\n& = \\underline{\\left( \\begin{array}{cc} a & -a^2 \\\\ 1 & -a \\end{array} \\right)}\r\n\\end{align}\\]\r\n

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

      \\(c = 1\\) \u3088\u308a, \\(A \\neq kE\\) \uff08 \\(k\\) \u306f\u5b9f\u6570\uff09\u306a\u306e\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n( a+d , ad-b ) & = ( 1 , 0 ) \\\\\r\n\\text{\u2234} \\quad d = 1-a , & \\ b = a(1-a)\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nA = \\left( \\begin{array}{cc} a & a(1-a) \\\\ 1 & 1-a \\end{array} \\right)\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nA^2 & = A , \\\\\r\nB^2 & = \\left( \\begin{array}{cc} a & -a^2 \\\\ 1 & -a \\end{array} \\right) \\left( \\begin{array}{cc} a & -a^2 \\\\ 1 & -a \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 0 & 0 \\\\ 0 & 0 \\end{array} \\right) = O , \\\\\r\nAB & = \\left( \\begin{array}{cc} a & a(1-a) \\\\ 1 & 1-a \\end{array} \\right) \\left( \\begin{array}{cc} a & -a^2 \\\\ 1 & -a \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} a & -a^2 \\\\ 1 & -a \\end{array} \\right) = A , \\\\\r\nBA & = \\left( \\begin{array}{cc} a & -a^2 \\\\ 1 & -a \\end{array} \\right) \\left( \\begin{array}{cc} a & a(1-a) \\\\ 1 & 1-a \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 0 & 0 \\\\ 0 & 0 \\end{array} \\right) = O\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u3066, \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066<\/p>\r\n

        \r\n
      1. [A] ... \u300c \\(( sA+tB )^n = x _ n A +y _ n B\\) \u3068\u8868\u305b\u308b. \u300d<\/li>\r\n<\/ol>\r\n

        \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u6570\u5b66\u7684\u5e30\u7d0d\u6cd5\u3092\u7528\u3044\u3066\u793a\u3059.<\/p>\r\n

          \r\n

        1. 1*<\/strong>\u3000\\(n = 1\\) \u306e\u3068\u304d, \\(sA+tB\\) \u306a\u306e\u3067\r\n\\[\r\nx _ 1 = s , \\ y _ 1 = t \\quad ... [4]\r\n\\]\r\n\u3068\u304a\u3051\u3070, [A] \u304c\u6210\u7acb\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n

        2. 2*<\/strong>\u3000\\(n = k\\) \uff08 \\(k\\) \u306f\u81ea\u7136\u6570\uff09\u306e\u3068\u304d\u306b, [A] \u304c\u6210\u7acb\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n( sA+tB )^{k+1} & = ( sA+tB ) ( x _ k A +y _ k B ) \\\\\r\n& = s x _ k A + s y _ k B\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\nx _ {k+1} = s x _ k , \\ y _ {k+1} = s y _ k \\quad ... [5]\r\n\\]\r\n\u3068\u304a\u3051\u3070, \\(n = k+1\\) \u306e\u3068\u304d\u3082, [A] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

          \u4ee5\u4e0a\u3088\u308a, \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066, [A]\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u304c\u793a\u3055\u308c\u305f.
          \r\n\u3055\u3089\u306b, [4] [5] \u3088\u308a\u6570\u5217 \\(\\{ x _ n \\}\\) \u306f, \u521d\u9805 \\(x _ 1 = s\\) , \u516c\u6bd4 \\(s\\) , \u6570\u5217 \\(\\{ y _ n \\}\\) \u306f, \u521d\u9805 \\(y _ 1 = t\\) , \u516c\u6bd4 \\(s\\) \u306e\u7b49\u6bd4\u6570\u5217\u306a\u306e\u3067\r\n\\[\r\nx _ n = \\underline{s^{n}} , \\ y _ n = \\underline{t s^{n-1}}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306f \\(A^2 = A\\) \u3092\u6e80\u305f\u3059. \u884c\u5217 \\(B\\) \u306f \\(B \\lef … \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":[104],"tags":[111,9],"class_list":["post-881","post","type-post","status-publish","format-standard","hentry","category-yokokoku_r_2013","tag-111","tag-yokokoku_r"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/881","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=881"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/881\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=881"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=881"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=881"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}