{"id":1303,"date":"2015-10-11T21:57:35","date_gmt":"2015-10-11T12:57:35","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1303"},"modified":"2021-09-29T22:47:09","modified_gmt":"2021-09-29T13:47:09","slug":"tbr201507","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201507\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2015\uff1a\u7b2c7\u554f"},"content":{"rendered":"
\n

\\(\\alpha , \\beta\\) \u306f\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5b9f\u6570\u3068\u3059\u308b.\r\n\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066 \\(2\\) \u70b9 \\(( \\alpha , 1 )\\) , \\(( \\beta , 1 )\\) \u3092\u305d\u308c\u305e\u308c\u70b9 \\(( {\\alpha}^2 , \\alpha )\\) , \\(( {\\beta}^2 , \\beta )\\) \u306b\u79fb\u3059 \\(1\\) \u6b21\u5909\u63db\u3092\u8868\u3059\u884c\u5217\u3092 \\(A\\) \u3068\u3059\u308b. \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057, \u70b9 \\(\\text{P} {} _ n \\ ( x _ n , y _ n )\\) \u3092\r\n\\[\r\n\\left( \\begin{array}{c} x _ 1 \\\\ y _ 1 \\end{array} \\right) = \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) , \\ \\left( \\begin{array}{c} x _ {n+1} \\\\ y _ {n+1} \\end{array} \\right) = A \\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) \\quad ( n = 1, 2, 3, \\cdots )\r\n\\]\r\n\u306b\u3088\u3063\u3066\u5b9a\u3081\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(Q = \\left( \\begin{array}{cc} \\alpha & \\beta \\\\ 1 & 1 \\end{array} \\right)\\) \u3068\u304a\u304f\u3068, \\(AQ = Q \\left( \\begin{array}{cc} \\alpha & 0 \\\\ 0 & \\beta \\end{array} \\right)\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u6570\u5217 \\(\\{ x _ n \\} , \\{ y _ n \\}\\) \u306e\u4e00\u822c\u9805\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u70b9 \\(\\text{P} {} _ 2 , \\text{P} {} _ 3 , \\text{P} {} _ 4 , \\cdots\\) \u304c\u3059\u3079\u3066\u76f4\u7dda \\(y = \\dfrac{1}{2} x\\) \u4e0a\u306b\u3042\u308b\u3088\u3046\u306a \\(\\alpha , \\beta\\) \u3092 \\(1\\) \u7d44\u6c42\u3081, \u305d\u306e\u3068\u304d\u306e\u884c\u5217 \\(A\\) \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

    \u6761\u4ef6\u3088\u308a\r\n\\[\r\nA \\left(\\begin{array}{cc} \\alpha & \\beta \\\\ 1 & 1 \\end{array}\\right) = AQ = \\left(\\begin{array}{cc} {\\alpha}^2 & {\\beta}^2 \\\\ \\alpha & \\beta \\end{array}\\right) \\quad ... [1]\r\n\\]\r\n\u307e\u305f\r\n\\[\r\nQ \\left(\\begin{array}{cc} \\alpha & 0 \\\\ 0 & \\beta \\end{array}\\right) = \\left(\\begin{array}{cc} \\alpha & \\beta \\\\ 1 & 1 \\end{array}\\right) \\left(\\begin{array}{cc} \\alpha & 0 \\\\ 0 & \\beta \\end{array}\\right) = \\left(\\begin{array}{cc} {\\alpha}^2 & {\\beta}^2 \\\\ \\alpha & \\beta \\end{array}\\right)\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nAQ = Q \\left( \\begin{array}{cc} \\alpha & 0 \\\\ 0 & \\beta \\end{array} \\right)\r\n\\]\r\n

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

    \u4e0e\u3048\u3089\u308c\u305f\u6f38\u5316\u5f0f\u3088\u308a\r\n\\[\r\n\\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) = A^{n-1} \\left( \\begin{array}{c} x _ 1 \\\\ y _ 1 \\end{array} \\right) = A^{n-1} \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right)\r\n\\]\r\n\u6761\u4ef6\u3088\u308a \\(|Q| = \\alpha -\\beta \\neq 0\\) \u306a\u306e\u3067, \\(Q\\) \u306b\u306f\u9006\u884c\u5217 \\(Q^{-1}\\) \u304c\u5b58\u5728\u3057\r\n\\[\r\nQ^{-1} = \\dfrac{1}{\\alpha -\\beta} \\left(\\begin{array}{cc} 1 & -\\beta \\\\ -1 & \\alpha \\end{array}\\right)\r\n\\]\r\n\u307e\u305f, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a \\(A = Q \\left( \\begin{array}{cc} \\alpha & 0 \\\\ 0 & \\beta \\end{array} \\right) Q^{-1}\\) \u306a\u306e\u3067, \\(n\\) \u56de\u639b\u3051\u3042\u308f\u305b\u308b\u3068\r\n\\[\r\nA^n = Q \\left( \\begin{array}{cc} {\\alpha}^n & 0 \\\\ 0 & {\\beta}^n \\end{array} \\right) Q^{-1}\r\n\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u308c\u3070, \\(n \\geqq 2\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{c} x _ n \\\\ y _ n \\end{array} \\right) & = Q \\left( \\begin{array}{cc} {\\alpha}^{n-1} & 0 \\\\ 0 & {\\beta}^{n-1} \\end{array} \\right) Q^{-1} \\left( \\begin{array}{c} 1 \\\\ 0 \\end{array} \\right) \\\\\r\n& = \\dfrac{1}{\\alpha -\\beta} \\left(\\begin{array}{cc} \\alpha & \\beta \\\\ 1 & 1 \\end{array}\\right) \\left( \\begin{array}{cc} {\\alpha}^{n-1} & 0 \\\\ 0 & {\\beta}^{n-1} \\end{array} \\right) \\left( \\begin{array}{c} 1 \\\\ -1 \\end{array} \\right) \\\\\r\n& = \\dfrac{1}{\\alpha -\\beta} \\left(\\begin{array}{cc} \\alpha & \\beta \\\\ 1 & 1 \\end{array}\\right) \\left( \\begin{array}{c} {\\alpha}^{n-1} \\\\ -{\\beta}^{n-1} \\end{array} \\right) \\\\\r\n& = \\dfrac{1}{\\alpha -\\beta} \\left( \\begin{array}{c} {\\alpha}^n -{\\beta}^n \\\\ {\\alpha}^{n-1} -{\\beta}^{n-1} \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nx _ n = \\underline{\\dfrac{{\\alpha}^n -{\\beta}^n}{\\alpha -\\beta}} , \\ y _ n = \\underline{\\dfrac{{\\alpha}^{n-1} -{\\beta}^{n-1}}{\\alpha -\\beta}}\r\n\\]\r\n\u3053\u308c\u306f, \\(n = 1\\) \u306e\u3068\u304d\u3082\u6e80\u305f\u3057\u3066\u3044\u308b.<\/p>\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u3092\u4ee3\u5165\u3059\u308c\u3070, \\(n \\geqq 2\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{{\\alpha}^{n-1} -{\\beta}^{n-1}}{\\alpha -\\beta} & = \\dfrac{1}{2} \\cdot \\dfrac{{\\alpha}^n -{\\beta}^n}{\\alpha -\\beta} \\\\\r\n\\text{\u2234} \\quad {\\alpha}^n -{\\beta}^n & = 2 ( {\\alpha}^{n-1} -{\\beta}^{n-1} )\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7e70\u308a\u8fd4\u3057\u7528\u3044\u308c\u3070\r\n\\[\r\n{\\alpha}^n -{\\beta}^n = 2^{n-1} ( \\alpha -\\beta )\r\n\\]\r\n\u3053\u308c\u306f, \\(n = 1\\) \u306e\u3068\u304d\u3082\u6e80\u305f\u3057\u3066\u3044\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\n{\\alpha}^2 -{\\beta}^2 & = 2 ( \\alpha -\\beta ) \\\\\r\n\\text{\u2234} \\quad \\alpha +\\beta & = 2 \\quad ... [2]\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n{\\alpha}^3 -{\\beta}^3 & = 4 ( \\alpha -\\beta ) \\\\\r\n( \\alpha +\\beta )^2 -\\alpha \\beta & = 4 \\\\\r\n\\text{\u2234} \\quad \\alpha \\beta & = 0 \\quad ... [3]\r\n\\end{align}\\]\r\n[2] [3] \u3088\u308a, \u6c42\u3081\u308b \\(1\\) \u7d44\u306f\r\n\\[\r\n( \\alpha , \\beta ) = \\underline{( 2 , 0 )}\r\n\\]\r\n\u3053\u306e\u3068\u304d, [1] \u3088\u308a\r\n\\[\\begin{align}\r\nA & = \\left(\\begin{array}{cc} {\\alpha}^2 & {\\beta}^2 \\\\ \\alpha & \\beta \\end{array}\\right) Q^{-1} \\\\\r\n& = \\dfrac{1}{\\alpha -\\beta} \\left(\\begin{array}{cc} {\\alpha}^2 -{\\beta}^2 & \\alpha \\beta ( \\alpha -\\beta ) \\\\ \\alpha -\\beta & 0 \\end{array}\\right) \\\\\r\n& = \\left(\\begin{array}{cc} \\alpha +\\beta & \\alpha \\beta \\\\ 1 & 0 \\end{array}\\right) \\\\\r\n& = \\underline{\\left(\\begin{array}{cc} 2 & 0 \\\\ 1 & 0 \\end{array}\\right)}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(\\alpha , \\beta\\) \u306f\u7570\u306a\u308b \\(2\\) \u3064\u306e\u5b9f\u6570\u3068\u3059\u308b. \u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066 \\(2\\) \u70b9 \\(( \\alpha , 1 )\\) , \\(( \\beta , 1 )\\) \u3092\u305d\u308c\u305e\u308c\u70b9 \\(( {\\al … \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":[132],"tags":[144,137],"class_list":["post-1303","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2015","tag-tsukuba_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1303","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=1303"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1303\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1303"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1303"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1303"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}