{"id":179,"date":"2011-12-02T22:39:24","date_gmt":"2011-12-02T13:39:24","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=179"},"modified":"2021-03-24T08:18:15","modified_gmt":"2021-03-23T23:18:15","slug":"kyr200904","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr200904\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb\u4e592009\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u3092 \\(ad-bd = 1\\) \u3092\u307f\u305f\u3059\u884c\u5217\u3068\u3059\u308b\uff08 \\(a , b , c , d\\) \u306f\u5b9f\u6570\uff09.\r\n\u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\u5e73\u9762\u4e0a\u306e\u70b9 \\(\\text{P} {} _ n \\, \\left( x _ n , y _ n \\right)\\) \u3092\r\n\\[\r\n\\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) = A^n \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right)\r\n\\]\r\n\u306b\u3088\u308a\u5b9a\u3081\u308b.\r\n\\(\\overrightarrow{\\text{OP} {} _ 1}\\) \u3068 \\(\\overrightarrow{\\text{OP} {} _ 2}\\) \u306e\u9577\u3055\u304c \\(1\\) \u306e\u3068\u304d,\r\n\u3059\u3079\u3066\u306e \\(n\\) \u306b\u5bfe\u3057\u3066 \\(\\overrightarrow{\\text{OP} {} _ n}\\) \u306e\u9577\u3055\u304c \\(1\\) \u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.\r\n\u3053\u3053\u3067 O \u306f\u539f\u70b9\u3067\u3042\u308b.<\/p>\r\n

\r\n\r\n

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

\\[\r\n\\overrightarrow{\\text{OP} {} _ 1} = A \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} a \\\\ c \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \u6761\u4ef6\u3088\u308a\r\n\\[\r\na^2+c^2=1 \\quad ... [1]\r\n\\]\r\n\u30cf\u30df\u30eb\u30c8\u30f3\u30fb\u30b1\u30fc\u30ea\u30fc\u306e\u5b9a\u7406\u3088\u308a,\r\n\\[\r\nA^2 = (a-d) A -E \\quad ... [2]\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\n\\overrightarrow{\\text{OP} {} _ 2} = \\left\\{ (a-d) A -E \\right\\} \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) = \\left( \\begin{array}{c} a(a-d)-1 \\\\ c(a-d) \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \u6761\u4ef6\u3088\u308a\r\n\\[\\begin{align}\r\n\\left\\{ a(a-d)-1 \\right\\}^2 +c^2(a-d)^2 & =1 \\\\\r\n(a-d)^2(a^2+c^2) -2a(a-d) & =0 \\\\\r\n\\text{\u2234} \\quad (a-d)(a+d) & =0 \\quad ( \\ \\text{\u2235} \\ [1]\n\\end{align}\\]\r\n

    \r\n

  1. 1*<\/strong>\u3000\\(a-d=0\\) \u306e\u3068\u304d
    \r\n[2]\u3088\u308a\r\n\\[\r\nA^2 =-E\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\left( \\begin{array}{c} x _ {n+2} \\\\ y _ {n+2} \\end{array} \\right) = \\left( \\begin{array}{c} -x _ n \\\\ -y _ n \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \\({x _ n}^2+{y _ n}^2=1\\) \u306a\u3089\u3070\r\n\\[\r\n{x _ {n+2}}^2 +{y _ {n+2}}^2 =1\r\n\\]\r\n\u3053\u308c\u3068, \u6761\u4ef6\u3088\u308a \\(\\text{OP} {} _ 1 =1\\) , \\(\\text{OP} {} _ 2 =1\\) \u306a\u306e\u3067,
    \r\n\u5e30\u7d0d\u7684\u306b\u3059\u3079\u3066\u306e \\(n\\) \u306b\u3064\u3044\u3066, \\(\\text{OP} {} _ n =1\\) .<\/p><\/li>\r\n

  2. 2*<\/strong>\u3000\\(a+d=0\\) \u306e\u3068\u304d
    \r\n\\(a-d=2a\\) \u306a\u306e\u3067, [2]\u3088\u308a\r\n\\[\r\nA^2 =2aA -E\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\left( \\begin{array}{c} x _ {n+2} \\\\ y _ {n+2} \\end{array} \\right) = \\left( \\begin{array}{c} 2ax _ {n+1}-x _ n \\\\ 2ay _ {n+1}-y _ n \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067, \\({x _ {n+1}}^2+{y _ {n+1}}^2=1\\) , \\({x _ n}^2+{y _ n}^2=1\\) \u306a\u3089\u3070\r\n\\[\\begin{align}\r\n{x _ {n+2}}^2 +{y _ {n+2}}^2 & = \\left( 2ax _ {n+1}-x _ n \\right)^2 +\\left( 2ay _ {n+1}-y _ n \\right)^2 \\\\\r\n& = 4a^2+1 -4a \\left( x _ {n+1} x _ n +y _ n y _ {n+1} \\right)\r\n\\end{align}\\]\r\n\u3055\u3089\u306b \\(x _ {n+1} x _ n +y _ n y _ {n+1} =a\\) \u3068\u4eee\u5b9a\u3059\u308c\u3070\r\n\\[\r\n{x _ {n+2}}^2 +{y _ {n+2}}^2 = 1\r\n\\]\r\n\u3053\u308c\u3068, \u6761\u4ef6\u3088\u308a \\(\\text{OP} {} _ 1 =1\\) , \\(\\text{OP} {} _ 2 =1\\) , \u3055\u3089\u306b\r\n\\[\\begin{align}\r\nx _ 1 x _ 2 +y _ 1 y _ 2 & = a(2a^2-1) +c \\cdot 2ac \\\\\r\n& = 2a \\left( a^2+c^2 \\right) -a =a \\quad ( \\ \\text{\u2235} \\ [1]\n\\end{align}\\]\r\n\u306a\u306e\u3067, \u5e30\u7d0d\u7684\u306b\u3059\u3079\u3066\u306e \\(n\\) \u306b\u3064\u3044\u3066, \\(\\text{OP} {} _ n =1\\) .<\/p><\/li>\r\n<\/ol>\r\n

    1*<\/strong> 2*<\/strong>\u3088\u308a, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u3092 \\(ad-bd = 1\\) \u3092\u307f\u305f\u3059\u884c\u5217\u3068\u3059\u308b\uff08 \\(a , b , c , d … \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":[21],"tags":[140,15],"class_list":["post-179","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2009","tag-kyoto_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/179","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=179"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/179\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=179"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=179"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=179"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}