{"id":657,"date":"2013-02-24T13:47:13","date_gmt":"2013-02-24T04:47:13","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=657"},"modified":"2021-09-10T19:12:20","modified_gmt":"2021-09-10T10:12:20","slug":"osr200705","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr200705\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2007\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. \\(4\\) \u500b\u306e\u884c\u5217\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) , \\quad B = \\left( \\begin{array}{ccc} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right) , \\\\\r\nC & = \\left( \\begin{array}{cc} 1 & -1 \\\\ -1 & 1 \\\\ 1 & -1 \\end{array} \\right) , \\quad D = \\left( \\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\u3092\u91cd\u8907\u3092\u8a31\u3057\u3066 \\(n\\) \u500b\u4e26\u3079\u305f\u3082\u306e\u3092\r\n\\[\r\nM _ 1 , M _ 2 , \\cdots , M _ n \\ .\r\n\\]\r\n\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u7a4d \\(M _ 1 M _ 2 \\cdots M _ n\\) \u304c\u5b9a\u7fa9\u3067\u304d\u308b\u5834\u5408\u306f\u4f55\u901a\u308a\u3042\u308b\u304b. \u305d\u306e\u6570\u3092 \\(n\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u7a4d \\(M _ 1 M _ 2 \\cdots M _ n\\) \u304c\u5b9a\u7fa9\u3067\u304d\u3066, \u305d\u306e\u7a4d\u304c\u96f6\u884c\u5217\u3067\u306a\u3044 \\(2 \\times 3\\) \u884c\u5217\u3068\u306a\u308b\u5834\u5408\u306f\u4f55\u901a\u308a\u3042\u308b\u304b. \u305d\u306e\u6570\u3092 \\(n\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u7a4d \\(M _ 1 M _ 2 \\cdots M _ n\\) \u304c\u5b9a\u7fa9\u3067\u304d\u3066, \u305d\u306e\u7a4d\u304c\u96f6\u884c\u5217\u3068\u306a\u3089\u306a\u3044\u5834\u5408\u306f\u4f55\u901a\u308a\u3042\u308b\u304b. \u305d\u306e\u6570\u3092 \\(n\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    \\(X _ n = M _ 1 M _ 2 \\cdots M _ n\\) \u3068\u304a\u304f.
    \r\n\\(k \\times \\ell\\) \u884c\u5217 \\(P\\) \u3068 \\(\\ell \\times m\\) \u884c\u5217 \\(Q\\) \u306b\u5bfe\u3057\u3066\u7a4d \\(PQ\\) \u304c\u5b9a\u7fa9\u3067\u304d\u3066, \u305d\u306e\u7a4d\u306f \\(k \\times m\\) \u884c\u5217\u306b\u306a\u308b.\r\n\u3057\u305f\u304c\u3063\u3066, \\(2 \\times 2\\) \u884c\u5217, \\(2 \\times 3\\) \u884c\u5217, \\(3 \\times 2\\) \u884c\u5217, \\(3 \\times 3\\) \u884c\u5217\u3069\u3046\u3057\u306e\u7a4d\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a\u3068\u306a\u308b\uff08\u7a4d\u304c\u5b9a\u7fa9\u3067\u304d\u306a\u3044\u5834\u6240\u306f\u00d7, \u3067\u304d\u308b\u5834\u6240\u306f\u7a4d\u306e\u884c\u5217\u306e\u5f62\uff09.\r\n\\[\r\n\\begin{array}{c|cccc} PQ & 2 \\times 2 & 2 \\times 3 & 3 \\times 2 & 3 \\times 3 \\\\ \\hline 2 \\times 2 & 2 \\times 2 & 2 \\times 3 & \\times & \\times \\\\ 2 \\times 3 & \\times & \\times & 2 \\times 2 & 2 \\times 3 \\\\ 3 \\times 2 & 3 \\times 2 & 3 \\times 3 & \\times & \\times \\\\ 3 \\times 3 & \\times & \\times & 3 \\times 2 & 3 \\times 3 \\end{array}\r\n\\]\r\n\u3053\u306e\u3053\u3068\u304b\u3089, \\(X _ n \\ ( n \\geqq 1 )\\) \u306f \\(2 \\times 2\\) \u884c\u5217, \\(2 \\times 3\\) \u884c\u5217, \\(3 \\times 2\\) \u884c\u5217, \\(3 \\times 3\\) \u884c\u5217\u306e\u3044\u305a\u308c\u304b\u3067\u3042\u308b.
    \r\n\u307e\u305f, \u4e0a\u306e\u8868\u304b\u3089, \\(X _ n\\) \u306b\u5bfe\u3057\u3066 \\(X _ {n+1}\\) \u3092\u5b9a\u7fa9\u3067\u304d\u308b\u3088\u3046\u306a \\(M _ {n+1}\\) \u306f \\(2\\) \u901a\u308a\u3042\u308b.
    \r\n\u3088\u3063\u3066, \\(X _ 1\\) \u306f \\(A , B , C , D\\) \u306e \\(4\\) \u901a\u308a\u3042\u308b\u306e\u3067, \\(X _ n\\) \u304c\u5b9a\u7fa9\u3067\u304d\u308b\u5834\u5408\u306e\u6570\u306f\r\n\\[\r\n4 \\cdot 2^{n-1} = \\underline{2^{n+1}} \\ .\r\n\\]\r\n

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

    \\(A , D\\) \u306f\u305d\u308c\u305e\u308c \\(2\\) \u6b21, \\(3\\) \u6b21\u5358\u4f4d\u884c\u5217\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nAA & = A , \\ CA = C , \\ AB = B , \\\\\r\nDC & = C , \\ BD = B , \\ DD = D \\ .\r\n\\end{align}\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\nBC & = \\left( \\begin{array}{ccc} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right) \\left( \\begin{array}{cc} 1 & -1 \\\\ -1 & 1 \\\\ 1 & -1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 0 & 0 \\\\ 0 & 0 \\end{array} \\right) = O , \\\\\r\nCB & = \\left( \\begin{array}{cc} 1 & -1 \\\\ -1 & 1 \\\\ 1 & -1 \\end{array} \\right) \\left( \\begin{array}{ccc} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{ccc} 1 & 0 & -1 \\\\ -1 & 0 & 1 \\\\ 1 & 0 & -1 \\end{array} \\right) \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u3092 \\(F\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nFD & = DF = F , \\\\\r\nFF & = \\left( \\begin{array}{ccc} 1 & 0 & -1 \\\\ -1 & 0 & 1 \\\\ 1 & 0 & -1 \\end{array} \\right) \\left( \\begin{array}{ccc} 1 & 0 & -1 \\\\ -1 & 0 & 1 \\\\ 1 & 0 & -1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{ccc} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right) = O , \\\\\r\nBF & = \\left( \\begin{array}{ccc} 1 & 1 & 0 \\\\ 0 & 1 & 1 \\end{array} \\right) \\left( \\begin{array}{ccc} 1 & 0 & -1 \\\\ -1 & 0 & 1 \\\\ 1 & 0 & -1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{ccc} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{array} \\right) = O , \\\\\r\nFC & = \\left( \\begin{array}{ccc} 1 & 0 & -1 \\\\ -1 & 0 & 1 \\\\ 1 & 0 & -1 \\end{array} \\right) \\left( \\begin{array}{cc} 1 & -1 \\\\ -1 & 1 \\\\ 1 & -1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 0 & 0 \\\\ 0 & 0 \\\\ 0 & 0 \\end{array} \\right) = O \\ .\r\n\\end{align}\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\(A , B , C , D , F\\) \u3069\u3046\u3057\u306e\u7a4d\u306f\u5b9a\u7fa9\u3055\u308c\u308b\u3068\u304d, \u4e0b\u8868\u306e\u3088\u3046\u306b\u96f6\u884c\u5217\u304b \\(A , B , C , D , F\\) \u306e\u3044\u305a\u308c\u304b\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} & A & B & C & D & F \\\\ \\hline A & A & B & \\times & \\times & \\times \\\\ B & \\times & \\times & O & B & O \\\\ C & C & F & \\times & \\times & \\times \\\\ D & \\times & \\times & C & D & F \\\\ F & \\times & \\times & O & F & O \\end{array} \\ .\r\n\\]\r\n\\(X _ n\\) \u304c\u96f6\u884c\u5217\u3067\u306f\u306a\u3044 \\(2 \\times 3\\) \u884c\u5217\u3068\u306a\u308b\u3068\u304d\u306f, \\(X _ n = B\\) \u3067\u3042\u308a, \u3053\u306e\u3088\u3046\u306b\u306a\u308b\u306e\u306f, \\(0 \\leqq k \\leqq n-1\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nM _ 1 = \\cdots = M _ k = A , \\ M _ {k+1} = \\cdots = M _ n = B \\ .\r\n\\]\r\n\u3067\u3042\u308b\u3068\u304d.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u5834\u5408\u306e\u6570\u306f\r\n\\[\r\n{} _ {n} \\text{C} {} _ {1} = \\underline{n} \\ .\r\n\\]\r\n

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

    \\(X _ n\\) \u306e\u7a2e\u985e\u306b\u3088\u3063\u3066, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(X _ n = A\\) \u306e\u3068\u304d\r\n\\[\r\nM _ 1 = \\cdots = M _ n = A \\ .\r\n\\]\r\n\u3067\u3042\u308b\u3068\u304d\u306b\u9650\u308b\u306e\u3067, \\(1\\) \u901a\u308a.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(X _ n = B\\) \u306e\u3068\u304d
      \r\n(2)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(n\\) \u901a\u308a.<\/p><\/li>\r\n

    3. 3*<\/strong>\u3000\\(X _ n = C\\) \u306e\u3068\u304d
      \r\n(2)<\/strong> \u306e\u3068\u304d\u3068\u540c\u69d8\u306b\u8003\u3048\u308c\u3070, \\(0 \\leqq k \\leqq n-1\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\nM _ 1 = \\cdots = M _ {k-1} = D , \\ M _ {k+1} = \\cdots = M _ {n} = A \\ .\r\n\\]\r\n\u3067\u3042\u308b\u3068\u304d\u306a\u306e\u3067, \\(n\\) \u901a\u308a.<\/p><\/li>\r\n

    4. 4*<\/strong>\u3000\\(X _ n = D\\) \u306e\u3068\u304d
      \r\n1*<\/strong> \u306e\u3068\u304d\u3068\u540c\u69d8\u306b\u8003\u3048\u3066,\r\n\\[\r\nM _ 1 = \\cdots = M _ n = D \\ .\r\n\\]\r\n\u3067\u3042\u308b\u3068\u304d\u306b\u9650\u308b\u306e\u3067, \\(1\\) \u901a\u308a.<\/p><\/li>\r\n

    5. 5*<\/strong>\u3000\\(X _ n = F\\) \u306e\u3068\u304d
      \r\n\\(1 \\leqq k \\lt \\ell \\leqq n\\) \u306b\u5bfe\u3057\u3066\r\n\\[\\begin{align}\r\nM _ 1 & = \\cdots = M _ {k-1} = D , \\quad M _ k = C , \\\\\r\nM _ {k+1} & = \\cdots = M _ {\\ell -1} = A , \\quad M _ {\\ell} = B , \\\\\r\nM _ {\\ell +1} & = \\cdots = M _ {n} = D \\ .\r\n\\end{align}\\]\r\n\u3067\u3042\u308b\u3068\u304d.
      \r\n\u3088\u3063\u3066, \\(C , B\\) \u306e\u9078\u3073\u65b9\u306b\u7740\u76ee\u3057\u3066,\r\n\\[\r\n{} _ {n} \\text{C} {} _ {2} = \\dfrac{n(n-1)}{2} \\ .\r\n\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u5834\u5408\u306e\u6570\u306f\r\n\\[\r\n1+n+n+1+\\dfrac{n(n-1)}{2} = \\underline{\\dfrac{n^2 +3n +4}{2}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(n\\) \u3092 \\(2\\) \u4ee5\u4e0a\u306e\u81ea\u7136\u6570\u3068\u3059\u308b. \\(4\\) \u500b\u306e\u884c\u5217 \\[\\begin{align} A & = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} … \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":[89],"tags":[142,109],"class_list":["post-657","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2007","tag-osaka_r","tag-109"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/657","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=657"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/657\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=657"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=657"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=657"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}