{"id":921,"date":"2013-09-04T21:39:59","date_gmt":"2013-09-04T12:39:59","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=921"},"modified":"2021-09-30T00:02:29","modified_gmt":"2021-09-29T15:02:29","slug":"kbr201305","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kbr201305\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2013\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.\r\n\u305f\u3060\u3057 \\(a , b , c , d\\) \u306f\u5b9f\u6570\u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(A^2 = \\left( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right)\\) \u3092\u6e80\u305f\u3059 \\(A\\) \u306f\u5b58\u5728\u3057\u306a\u3044\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(A^2 = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right)\\) \u3092\u6e80\u305f\u3059 \\(A\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u3067\u6c42\u3081\u305f \\(A\\) \u306e\u305d\u308c\u305e\u308c\u306b\u3064\u3044\u3066 \\(A+A^2+A^3+ \\cdots +A^{2013}\\) \u3092\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\u3092\u307f\u305f\u3059\u884c\u5217 \\(A\\) \u304c\u5b58\u5728\u3059\u308b\u3068\u4eee\u5b9a\u3059\u308b\u3068\r\n\\[\r\nA^2 = \\left( \\begin{array}{cc} a^2+bc & b(a+d) \\\\ c(a+d) & d^2+bc \\end{array} \\right) = \\left( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\left\\{\\begin{array}{ll} a^2+bc = d^2+bc = 0 & ... [1] \\\\ b(a+d) = c(a+d) = 1 & ... [2] \\end{array}\\right.\r\n\\]\r\n[2] \u3088\u308a, \\(a+d \\neq 0\\) ... [3] \u3067\u3042\u308a\r\n\\[\r\nb = c \\neq 0 \\quad ... [4]\r\n\\]\r\n\u3053\u308c\u3092 [1] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{gather}\r\na^2 +b^2 = d^2 +b^2 = 0 \\\\\r\n\\text{\u2234} \\quad a = d = b = c = 0\r\n\\end{gather}\\]\r\n\u3053\u308c\u306f, [3] [4] \u306b\u77db\u76fe\u3059\u308b.
    \r\n\u3088\u3063\u3066, \\(A^2 = \\left( \\begin{array}{cc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right)\\) \u3092\u307f\u305f\u3059 \\(A\\) \u306f\u5b58\u5728\u3057\u306a\u3044.<\/p>\r\n

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

    \u6761\u4ef6\u3088\u308a\r\n\\[\r\nA^2 = \\left( \\begin{array}{cc} a^2+bc & b(a+d) \\\\ c(a+d) & d^2+bc \\end{array} \\right) = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right)\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\left\\{\\begin{array}{ll} a^2+bc = d^2+bc = 0 & ... [5] \\\\ b(a+d) = 1 & ... [6] \\\\ c(a+d) = -1 & ... [7] \\end{array}\\right.\r\n\\]\r\n[6] [7] \u3088\u308a, \\(a+d \\neq 0\\) ... [8] \u3067\r\n\\[\r\nb = -c = \\dfrac{1}{a+d} \\quad ... [9]\r\n\\]\r\n[5] \u3088\u308a\r\n\\[\\begin{align}\r\na^2 & = d^2 \\\\\r\n\\text{\u2234} \\quad d & = a \\quad ( \\ \\text{\u2235} \\ [8] \\ ) \\quad ... [10]\r\n\\end{align}\\]\r\n[9] [10] \u3092 [5] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{gather}\r\na^2 -\\left( \\dfrac{1}{2a} \\right)^2 = 0 \\\\\r\na^4 = \\dfrac{1}{4} \\\\\r\n\\text{\u2234} \\quad a = \\pm \\dfrac{1}{\\sqrt{2}}\r\n\\end{gather}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u884c\u5217 \\(A\\) \u306f\r\n\\[\r\nA = \\underline{\\pm \\dfrac{1}{\\sqrt{2}} \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 1 \\end{array} \\right)}\r\n\\]\r\n

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

    \\[\\begin{align}\r\nA^4 & = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) = -E \\\\\r\nA^8 & = (-E)^2 = E\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\textstyle\\sum\\limits _ {k=1}^8 A^k & = A+A^2+A^3-E-A-A^2-A^3+E \\\\\r\n& = O\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u81ea\u7136\u6570 \\(n\\) \u306b\u5bfe\u3057\u3066\r\n\\[\r\n\\textstyle\\sum\\limits _ {k=n}^{n+8} A^k = O\r\n\\]\r\n\u304c\u6210\u308a\u7acb\u3064.
    \r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, \\(2013 = 8 \\cdot 251 + 5\\) \u306a\u306e\u3067, \u6c42\u3081\u308b\u548c \\(S\\) \u306f\r\n\\[\r\nS = \\textstyle\\sum\\limits _ {k=1}^5 A^k = A^2+A^3-E\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\nA^3 & = \\pm \\dfrac{1}{\\sqrt{2}} \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 1 \\end{array} \\right) \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) \\\\\r\n& = \\pm \\dfrac{1}{\\sqrt{2}} \\left( \\begin{array}{cc} -1 & 1 \\\\ -1 & -1 \\end{array} \\right)\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nS & = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array} \\right) \\pm \\dfrac{1}{\\sqrt{2}} \\left( \\begin{array}{cc} -1 & 1 \\\\ -1 & -1 \\end{array} \\right) -\\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right) \\\\\r\n& = \\underline{\\left( 1 \\pm \\dfrac{1}{\\sqrt{2}} \\right) \\left( \\begin{array}{cc} -1 & 1 \\\\ -1 & -1 \\end{array} \\right)}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. \u305f\u3060\u3057 \\(a , b , … \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":[102],"tags":[144,111],"class_list":["post-921","post","type-post","status-publish","format-standard","hentry","category-tsukuba_r_2013","tag-tsukuba_r","tag-111"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/921","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=921"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/921\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=921"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=921"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=921"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}