{"id":164,"date":"2011-12-02T22:09:02","date_gmt":"2011-12-02T13:09:02","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=164"},"modified":"2021-03-16T16:13:43","modified_gmt":"2021-03-16T07:13:43","slug":"tkr200902","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr200902\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2009\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\u5b9f\u6570\u3092\u6210\u5206\u306b\u3082\u3064\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u3068\u5b9f\u6570 \\(r , s\\) \u304c\r\n\u4e0b\u306e\u6761\u4ef6 (i) , (ii) , (iii) \u3092\u307f\u305f\u3059\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (i)\u3000\\(s \\gt 1\\)<\/p><\/li>\r\n

  2. (ii)\u3000\\(A \\left( \\begin{array}{c} r \\\\ 1 \\end{array} \\right) = s \\left( \\begin{array}{c} r \\\\ 1 \\end{array} \\right)\\)<\/p><\/li>\r\n

  3. (iii)\u3000\\(A^n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right)\\) \uff08 \\(n= 1, 2, \\cdots\\) \uff09\u3068\u3059\u308b\u3068\u304d, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} x _ n = \\displaystyle\\lim _ {n \\rightarrow \\infty} y _ n = 0\\)<\/p><\/li>\r\n<\/ol>\r\n

    \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\\(B = \\left( \\begin{array}{cc} 1 & r \\\\ 0 & 1 \\end{array} \\right)^{-1} A \\left( \\begin{array}{cc} 1 & r \\\\ 0 & 1 \\end{array} \\right)\\) \u3092 \\(a , c , r , s\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\\(B^n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} z _ n \\\\ w _ n \\end{array} \\right)\\) \uff08 \\(n= 1, 2, \\cdots\\) \uff09\u3068\u3059\u308b\u3068\u304d, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} z _ n = \\displaystyle\\lim _ {n \\rightarrow \\infty} w _ n = 0\\) \u3092\u793a\u305b.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000\\(c=0\\) \u304b\u3064 \\(| a | \\lt 1\\) \u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

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

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

      \\(A \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} a \\\\ c \\end{array} \\right)\\) \u3068, \u6761\u4ef6(ii)\u3088\u308a\r\n\\[\r\nA \\left( \\begin{array}{cc} 1 & r \\\\ 0 & 1 \\end{array} \\right) = \\left( \\begin{array}{cc} a & c \\\\ sr & r \\end{array} \\right)\r\n\\]\r\n\u3053\u308c\u3068, \\(\\left( \\begin{array}{cc} 1 & r \\\\ 0 & 1 \\end{array} \\right)^{-1} = \\left( \\begin{array}{cc} 1 & 0 \\\\ r & 1 \\end{array} \\right)\\) \u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nB & = \\left( \\begin{array}{cc} 1 & 0 \\\\ r & 1 \\end{array} \\right) \\left( \\begin{array}{cc} a & c \\\\ sr & r \\end{array} \\right) \\\\\r\n& = \\underline{\\left( \\begin{array}{cc} a-rc & c \\\\ 0 & s \\end{array} \\right)}\r\n\\end{align}\\]\r\n

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

      \\(B^n = \\left( \\begin{array}{cc} 1 & 0 \\\\ r & 1 \\end{array} \\right) A^n \\left( \\begin{array}{cc} 1 & r \\\\ 0 & 1 \\end{array} \\right)\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nB^n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) & = \\left( \\begin{array}{cc} 1 & 0 \\\\ r & 1 \\end{array} \\right) A^n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 1 & 0 \\\\ r & 1 \\end{array} \\right) \\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{c} x _ n -r y _ n \\\\ y _ n \\end{array} \\right) = \\left( \\begin{array}{c} z _ n \\\\ w _ n \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3053\u308c\u3068, \u6761\u4ef6 (iii) \u3088\u308a\r\n\\[\r\n\\underline{\\displaystyle\\lim _ {n \\rightarrow \\infty} z _ n = \\displaystyle\\lim _ {n \\rightarrow \\infty} w _ n = 0}\r\n\\]\r\n

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

      \\(B^n = \\left( \\begin{array}{cc} a _ n & 0 \\\\ c _ n & d _ n \\end{array} \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\left( \\begin{array}{c} z _ n \\\\ w _ n \\end{array} \\right) = B^n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} a _ n \\\\ c _ n \\end{array} \\right)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} a _ n & = 0 \\quad ... [1] , \\\\\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} c _ n & = 0 \\quad ... [2]\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nB^{n+1} & = \\left( \\begin{array}{cc} a-rc & c \\\\ 0 & s \\end{array} \\right) \\left( \\begin{array}{cc} a _ n & 0 \\\\ c _ n & d _ n \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} (a-rc) a _ n & 0 \\\\ (a-rc)c _ n +cd _ n & sd _ n \\end{array} \\right) = \\left( \\begin{array}{cc} a _ {n+1} & 0 \\\\ c _ {n+1} & d _ {n+1} \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad & \\left\\{ \\begin{array}{ll} a _ {n+1} =(a-rc)a _ n & ...[3] \\\\ c _ {n+1} =(a-rc)c _ n +cd _ n & ...[4] \\\\ d _ {n+1} =sd _ n & ...[5] \\end{array} \\right.\r\n\\end{align}\\]\r\n\\(a _ 1 = a-rc\\) , \\(d _ 1 =s\\) \u306a\u306e\u3067, [3] [5]\u3088\u308a\r\n\\[\r\na _ n = (a-rc)^n , \\quad d _ n = s^n\r\n\\]\r\n\u6761\u4ef6 (i) \u3088\u308a, \\(\\displaystyle\\lim _ {n \\rightarrow \\infty} d _ n = \\infty\\) \u3067\u3042\u308b\u304c, [2] [4] \u3088\u308a\r\n\\[\r\n\\underline{c=0}\r\n\\]\r\n\u3053\u306e\u3068\u304d, \\(a _ n =a^n\\) \u3067\u3042\u308a, [1] \u3088\u308a\r\n\\[\r\n\\underline{| a | \\lt 1}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570\u3092\u6210\u5206\u306b\u3082\u3064\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u3068\u5b9f\u6570 \\(r , s\\) \u304c \u4e0b\u306e\u6761\u4ef6 (i) , (ii) … \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":[18],"tags":[139,15],"class_list":["post-164","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2009","tag-tokyo_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/164","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=164"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/164\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=164"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=164"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=164"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}