{"id":235,"date":"2011-12-03T00:32:33","date_gmt":"2011-12-02T15:32:33","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=235"},"modified":"2021-10-03T20:31:00","modified_gmt":"2021-10-03T11:31:00","slug":"kbr200905","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr200905\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2009\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057, \u884c\u5217 \\(A = \\left( \\begin{array}{cc} a-1 & a-2 \\\\ a-2 & 1-a \\end{array} \\right)\\) \u3092\u8003\u3048\u308b.\r\n\\(n\\) \u3092\u81ea\u7136\u6570\u3068\u3057, \u5ea7\u6a19\u5e73\u9762\u4e0a\u306b\u304a\u3044\u3066, \u884c\u5217 \\(A^n\\) \u306b\u3088\u308a\u70b9 \\(( 1, 0 )\\) \u304c\u70b9 \\(\\text{P} {} _ n\\) \u306b\u79fb\u308a, \u70b9 \\(( 0, 1 )\\) \u304c\u70b9 \\(\\text{Q} {} _ n\\) \u306b\u79fb\u308b\u3082\u306e\u3068\u3059\u308b. \\(2\\) \u70b9 \\(\\text{P} {} _ n , \\text{Q} {} _ n\\) \u306e\u9593\u306e\u8ddd\u96e2\u3092 \\(\\text{P} {} _ n \\text{Q} {} _ n\\) \u3067\u8868\u3059.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(\\text{P} {} _ 1 \\text{Q} {} _ 1\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(A^n\\) \u3092 \\(a\\) \u3068 \\(n\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(n\\) \u304c\u56fa\u5b9a\u3055\u308c, \\(a\\) \u304c\u5b9f\u6570\u5168\u4f53\u3092\u52d5\u304f\u3068\u304d, \\(\\text{P} {} _ n \\text{Q} {} _ n\\) \u306e\u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/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}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) = A\\) \u306a\u306e\u3067\r\n\\[\r\n\\overrightarrow{\\text{OP} {} _ 1} = \\left( \\begin{array}{c} a-1 \\\\ a-2 \\end{array} \\right) , \\ \\overrightarrow{\\text{OQ} {} _ 1} = \\left( \\begin{array}{c} a-2 \\\\ 1-a \\end{array} \\right)\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\text{P} {} _ 1 \\text{Q} {} _ 1 & = \\sqrt{1^2+(2a-3)^2} = \\sqrt{4a^2-12a+10} \\\\\r\n& = \\underline{\\sqrt{2 \\left( 2a^2-6a+5 \\right)}}\r\n\\end{align}\\]\r\n

    (2)<\/strong>\r\n\\[\\begin{align}\r\n\\det (A) & = -(a-1)^2 -(a-2)^2 = -\\left( 2a^2-6a+5 \\right) \\\\\r\n\\text{trace} (A) & = (a-1) -(a-1) = 0\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u30cf\u30df\u30eb\u30c8\u30f3\u30fb\u30b1\u30fc\u30ea\u30fc\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\\begin{gather}\r\nA^2 -\\left( 2a^2-6a+5 \\right) E = O \\\\\r\n\\text{\u2234} \\quad A^2 = \\left( 2a^2-6a+5 \\right) E\r\n\\end{gather}\\]\r\n\u3088\u3063\u3066, \u3053\u308c\u3092\u7e70\u308a\u8fd4\u3057\u7528\u3044\u308c\u3070, \\(n\\) \u306e\u5076\u5947\u3067\u5834\u5408\u5206\u3051\u3057\u3066\r\n\\[\r\nA^n = \\underline{\\left\\{ \\begin{array}{ll} \\left( 2a^2-6a+5 \\right)^{\\frac{n}{2}} \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) & \\ ( \\ n \\ \\text{\u304c\u5076\u6570\u306e\u3068\u304d} ) \\\\ \\left( 2a^2-6a+5 \\right)^{\\frac{n-1}{2}} \\left( \\begin{array}{cc} a-1 & a-2 \\\\ a-2 & 1-a \\end{array} \\right) & \\ ( \\ n \\ \\text{\u304c\u5947\u6570\u306e\u3068\u304d} ) \\end{array} \\right.}\r\n\\]\r\n

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

      \r\n

    1. 1*<\/strong>\u3000\\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OP} {} _ n} & = \\left( 2a^2-6a+5 \\right)^{\\frac{n}{2}} \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) , \\\\\r\n\\overrightarrow{\\text{OQ} {} _ n} & = \\left( 2a^2-6a+5 \\right)^{\\frac{n}{2}} \\left( \\begin{array}{c} 0 \\\\ 1 \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad \\text{P} {} _ n\\text{Q} {} _ n & = \\sqrt{2} \\left( 2a^2-6a+5 \\right)^\\frac{n}{2} \\\\\r\n& = \\sqrt{2 \\left( 2a^2-6a+5 \\right)^n}\r\n\\end{align}\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(n\\) \u304c\u5947\u6570\u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OP} {} _ n} & = \\left( 2a^2-6a+5 \\right)^{\\frac{n-1}{2}} \\left( \\begin{array}{c} a-1 \\\\ a-2 \\end{array} \\right) , \\\\\r\n\\overrightarrow{\\text{OQ} {} _ n} & = \\left( 2a^2-6a+5 \\right)^{\\frac{n-1}{2}} \\left( \\begin{array}{c} a-2 \\\\ 1-a \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad \\text{P} {} _ n\\text{Q} {} _ n & = \\sqrt{2 \\left( 2a^2-6a+5 \\right)} \\left( 2a^2-6a+5 \\right)^\\frac{n-1}{2} \\\\\r\n& = \\sqrt{2 \\left( 2a^2-6a+5 \\right)^n}\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      1*<\/strong> 2*<\/strong>\u3088\u308a \\(n\\) \u306e\u5076\u5947\u306b\u3088\u3089\u305a\r\n\\[\\begin{align}\r\n\\text{P} {} _ n\\text{Q} {} _ n & = \\sqrt{2 \\left( 2a^2-6a+5 \\right)^n} \\\\\r\n& = \\sqrt{2 \\left\\{ 2 \\left( a -\\dfrac{3}{2} \\right)^2 +\\dfrac{1}{2} \\right\\}^n} \\\\\r\n& \\geqq \\sqrt{2 \\left( \\dfrac{1}{2} \\right)^n} \\\\\r\n& = 2^{\\frac{1-n}{2}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \\(a=\\dfrac{3}{2}\\) \u306e\u3068\u304d, \u6700\u5c0f\u5024 \\(\\underline{2^{\\frac{1-n}{2}}}\\) \u3092\u3068\u308b.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057, \u884c\u5217 \\(A = \\left( \\begin{array}{cc} a-1 & a-2 \\\\ a-2 & 1-a \\end{array} \\right)\\) \u3092\u8003\u3048\u308b. \\(n\\) \u3092\u81ea\u7136\u6570\u3068 … \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":[39],"tags":[144,15],"class_list":["post-235","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2009","tag-tsukuba_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/235","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=235"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/235\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=235"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=235"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}