{"id":839,"date":"2013-07-09T01:16:18","date_gmt":"2013-07-08T16:16:18","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=839"},"modified":"2021-09-14T08:28:51","modified_gmt":"2021-09-13T23:28:51","slug":"iks201302","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201302\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272013\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(2\\) \u6b21\u6b63\u65b9\u884c\u5217 \\(\\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306e\u3046\u3061, \u6b21\u306e \\(3\\) \u6761\u4ef6 (i) , (ii) , (iii) \u3092\u6e80\u305f\u3059\u3082\u306e\u5168\u4f53\u306e\u96c6\u5408\u3092 \\(M\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (i)\u3000\\(a , b , c , d\\) \u306f\u3059\u3079\u3066\u6574\u6570<\/p><\/li>\r\n

  2. (ii)\u3000\\(b+c = 0\\)<\/p><\/li>\r\n

  3. (iii)\u3000\\(a-b-d = 0\\)<\/p><\/li>\r\n<\/ol>\r\n

    \u307e\u305f \\(E\\) \u3092 \\(2\\) \u6b21\u5358\u4f4d\u884c\u5217\u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\u884c\u5217 \\(A , B\\) \u304c\u3068\u3082\u306b \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u308b\u3068\u304d, \u305d\u308c\u3089\u306e\u7a4d \\(AB\\) \u3082 \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u3068\u305d\u306e\u9006\u884c\u5217 \\(A^{-1}\\) \u304c\u3068\u3082\u306b \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u308b\u3068\u304d, \\(ad-bc = 1\\) \u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000\u884c\u5217 \\(A\\) \u3068\u305d\u306e\u9006\u884c\u5217 \\(A^{-1}\\) \u304c\u3068\u3082\u306b \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u308b\u3088\u3046\u306a \\(A\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n

    4. (4)<\/strong>\u3000\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u3063\u3066 \\(A^n = E\\) \u3092\u6e80\u305f\u3059\u3088\u3046\u306a\u884c\u5217 \\(A\\) \u306e\u5168\u4f53\u306e\u96c6\u5408\u3092 \\(S _ n\\) \u3068\u3059\u308b. \\(S _ n\\) \u306e\u8981\u7d20\u306e\u500b\u6570\u304c\u3061\u3087\u3046\u3069 \\(3\\) \u3068\u306a\u308b \\(n\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/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 (ii) , (iii) \u3088\u308a\r\n\\[\r\nc = -b , \\ d = a-b \\ .\r\n\\]\r\n\u306a\u306e\u3067, \u96c6\u5408 \\(M\\) \u306b\u5c5e\u3059\u308b\u884c\u5217\u306f,\r\n\\[\r\n\\left( \\begin{array}{cc} a & b \\\\ -b & a-b \\end{array} \\right) \\quad ( \\ a , b \\text{\u306f\u6574\u6570} ) \\quad ... [1] \\ .\r\n\\]\r\n\u3068\u8868\u305b\u308b.\r\n\\(A = \\left( \\begin{array}{cc} p & q \\\\ -q & p-q \\end{array} \\right)\\) , \\(B = \\left( \\begin{array}{cc} r & s \\\\ -s & r-s \\end{array} \\right)\\) \uff08 \\(p , q , r , s\\) \u306f\u6574\u6570\uff09\u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nAB & = \\left( \\begin{array}{cc} p & q \\\\ -q & p-q \\end{array} \\right) \\left( \\begin{array}{cc} r & s \\\\ -s & r-s \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} pr-qs & ps+q(r-s) \\\\ -ps-q(r-s) & -qs+(p-q)(r-s) \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} pr-qs & ps+q(r-s) \\\\ -ps-q(r-s) & (pr-qs) -\\left\\{ ps+q(r-s) \\right\\} \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\\(pr-qs , (ps+q(r-s)\\) \u3082\u6574\u6570\u306a\u306e\u3067, \\(AB\\) \u3082 [1] \u306e\u3088\u3046\u306b\u8868\u305b\u3066, \u96c6\u5408 \\(M\\) \u306b\u5c5e\u3059\u308b.<\/p>\r\n

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

      \u884c\u5217 \\(A\\) \u306e\u884c\u5217\u5f0f\u3092 \\(|A|\\) \u3068\u8868\u3059.
      \r\n\u9006\u884c\u5217 \\(A^{-1}\\) \u306b\u3064\u3044\u3066\r\n\\[\r\nA^{-1} = \\dfrac{1}{|A|} \\left( \\begin{array}{cc} d & -b \\\\ -c & a \\end{array} \\right) \\ .\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n| A^{-1} | = \\dfrac{ad-bc}{|A|^2} & = \\dfrac{1}{|A|} \\\\\r\n\\text{\u2234} \\quad |A| | A^{-1} | & = 1 \\ .\r\n\\end{align}\\]\r\n\\(A , A^{-1}\\) \u306f\u3068\u3082\u306b \\(M\\) \u306e\u8981\u7d20\u306a\u306e\u3067, \u305d\u306e\u884c\u5217\u5f0f \\(|A| , | A^{-1} |\\) \u306f\u3068\u3082\u306b\u6574\u6570\u3067\u3042\u308b\u304b\u3089\r\n\\[\r\n|A| = \\pm 1 \\ .\r\n\\]\r\n\u3068\u3053\u308d\u3067, [1] \u3088\u308a\r\n\\[\\begin{align}\r\n|A| & = a(a-b) -b(-b) \\\\\r\n& = a^2-ab+b^2 \\\\\r\n& = \\left( a-\\dfrac{b}{2} \\right)^2 +\\dfrac{3 b^2}{4} \\geqq 0 \\quad ... [2] \\ .\r\n\\end{align}\\]\r\n\u3067\u3042\u308b\u304b\u3089\r\n\\[\r\n|A| = ad-bc = 1 \\ .\r\n\\]\r\n

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

      (2)<\/strong> \u306e\u7d50\u679c\u3068, [2] \u3088\u308a\r\n\\[\\begin{align}\r\n\\left( a-\\dfrac{b}{2} \\right)^2 +\\dfrac{3 b^2}{4} & = 1 \\\\\r\n\\text{\u2234} \\quad (2a-b)^2+3b^2 & = 4 \\ .\r\n\\end{align}\\]\r\n\\(a , b\\) \u306f\u6574\u6570\u306a\u306e\u3067,\r\n\\[\\begin{align}\r\n( 2a-b , b ) & = ( \\pm 2 , 0 ) , ( \\pm 1 , \\pm 1 ) , ( \\pm 1 , \\mp 1 ) \\quad ( \\text{\u8907\u53f7\u540c\u9806} ) \\\\\r\n\\text{\u2234} \\quad (a,b) & = ( \\pm 1 , 0 ) , ( \\pm 1 , \\pm 1 ) , ( 0 , \\mp 1 ) \\quad ( \\text{\u8907\u53f7\u540c\u9806} ) \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u884c\u5217 \\(A\\) \u306f\r\n\\[\r\nA = \\underline{\\left( \\begin{array}{cc} \\pm 1 & 0 \\\\ 0 & \\pm 1 \\end{array} \\right) , \\left( \\begin{array}{cc} \\pm 1 & \\pm 1 \\\\ \\mp 1 & 0 \\end{array} \\right) , \\left( \\begin{array}{cc} 0 & \\mp 1 \\\\ \\pm 1 & \\pm 1 \\end{array} \\right)} \\quad ( \\text{\u8907\u53f7\u540c\u9806} ) \\ .\r\n\\]\r\n

      (4)<\/strong>\r\n\\[\r\nA^n = E \\ ... [3] \\ .\r\n\\]\r\n\u96c6\u5408 \\(S _ n\\) \u306e\u8981\u7d20\u306e\u500b\u6570\u3092 \\(|S _ n|\\) \u3067\u8868\u3059.
      \r\n\\(n=1\\) \u306e\u3068\u304d, \\(A=E\\) \u3067, \\(|S _ 1| = 1\\) \u306a\u306e\u3067, \u6761\u4ef6\u3092\u6e80\u305f\u3055\u306a\u3044.
      \r\n\u4ee5\u4e0b\u3067\u306f, \\(n \\geqq 2\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308b.
      \r\n(1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(A^m\\) \uff08 \\(m\\) \u306f\u4efb\u610f\u306e\u81ea\u7136\u6570\uff09\u306f \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u308b.
      \r\n\u307e\u305f, [3] \u3088\u308a\r\n\\[\r\nA^{-1} = A^{n-1} \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(A^{-1}\\) \u3082 \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u308b.
      \r\n\u3057\u305f\u304c\u3063\u3066, \\(A , A^{-1}\\) \u304c\u3068\u3082\u306b \\(M\\) \u306e\u8981\u7d20\u3067\u3042\u308b\u3088\u3046\u306a \\(A\\) \u306f, (3)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\r\nA = \\pm E , \\pm C ,\\pm D \\\\\r\n\\left( \\text{\u305f\u3060\u3057, } \\ C = \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 0 \\end{array} \\right) , \\ D = \\left( \\begin{array}{cc} 0 & -1 \\\\ 1 & 1 \\end{array} \\right) \\ \\right)\r\n\\]\r\n\u306e \\(6\\) \u3064\u304c\u8003\u3048\u3089\u308c\u308b.<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(A = E\\) \u306e\u3068\u304d
        \r\n\u3059\u3079\u3066\u306e\u81ea\u7136\u6570 \\(n\\) \u306b\u3064\u3044\u3066, [3] \u306f\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      2. 2*<\/strong>\u3000\\(A = -E\\) \u306e\u3068\u304d
        \r\n\\(A^2 = E\\) \u306a\u306e\u3067, \\(n\\) \u304c\u5076\u6570\u306e\u3068\u304d, [3] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      3. 3*<\/strong>\u3000\\(A = C\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nC^2 & = \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 0 \\end{array} \\right) \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 0 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & -1 \\end{array} \\right) = D \\quad ... [4] \\\\\r\nC^3 & = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & -1 \\end{array} \\right) \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 0 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) = E \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(n\\) \u304c \\(3\\) \u306e\u500d\u6570\u306e\u3068\u304d, [3] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      4. 4*<\/strong>\u3000\\(A = -C\\) \u306e\u3068\u304d
        \r\n3*<\/strong> \u306e\u3068\u304d\u3088\u308a\r\n\\[\r\nC^2 = D , \\ C^3 = -E \\ .\r\n\\]\r\n\u3055\u3089\u306b\r\n\\[\r\nC^4 = C , \\ C^5 = -D , \\ C^6 = E \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(n\\) \u304c \\(6\\) \u306e\u500d\u6570\u306e\u3068\u304d, [3] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      5. 5*<\/strong>\u3000\\(A=D\\) \u306e\u3068\u304d
        \r\n[4] \u3092\u7528\u3044\u308c\u3070\r\n\\[\r\nD^2 = C^4 = C , \\ D^3 = CD = E \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(n\\) \u304c \\(3\\) \u306e\u500d\u6570\u306e\u3068\u304d, [3] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n

      6. 6*<\/strong>\u3000\\(A=-D\\) \u306e\u3068\u304d
        \r\n5*<\/strong> \u306e\u3068\u304d\u3088\u308a\r\n\\[\r\nD^2 = C , \\ D^3 = -CD = -E \\ .\r\n\\]\r\n\u3055\u3089\u306b\r\n\\[\r\nD^4 = D , \\ D^5 = -C , \\ D^6 = E \\ .\r\n\\]\r\n\u306a\u306e\u3067, \\(n\\) \u304c \\(6\\) \u306e\u500d\u6570\u306e\u3068\u304d, [3] \u304c\u6210\u7acb\u3059\u308b.<\/p><\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \\(n\\) \u306e \\(\\text{mod} \\ 6\\) \u306e\u5270\u4f59\u3067\u5206\u3051\u3066\u8003\u3048\u308b\u3068\r\n\\[\r\n| S _ n | = \\left\\{\\begin{array}{ll} 6 & ( n \\equiv 0 \\text{\u306e\u3068\u304d} ) \\\\ 1 & ( n \\equiv \\pm 1 \\text{\u306e\u3068\u304d} ) \\\\ 2 & ( n \\equiv \\pm 2 \\text{\u306e\u3068\u304d} ) \\\\ 3 & ( n \\equiv 3 \\text{\u306e\u3068\u304d} ) \\end{array} \\right. \\ .\r\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b \\(n\\) \u306f\r\n\\[\r\nn = \\underline{6k-3} \\quad ( \\ k \\text{\u306f\u81ea\u7136\u6570} ) \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(2\\) \u6b21\u6b63\u65b9\u884c\u5217 \\(\\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306e\u3046\u3061, \u6b21\u306e \\(3\\) \u6761\u4ef6 (i) , (ii) , (iii … \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":[94],"tags":[145,111],"class_list":["post-839","post","type-post","status-publish","format-standard","hentry","category-ikashika_2013","tag-ikashika","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/839","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=839"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/839\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=839"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=839"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=839"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}