{"id":1180,"date":"2015-07-21T23:54:33","date_gmt":"2015-07-21T14:54:33","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1180"},"modified":"2021-09-29T23:07:21","modified_gmt":"2021-09-29T14:07:21","slug":"tbr201405","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tbr201405\/","title":{"rendered":"\u7b51\u6ce2\u5927\u7406\u7cfb2014\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\u5b9f\u6570\u3092\u6210\u5206\u3068\u3059\u308b\u6b63\u65b9\u884c\u5217\r\n\\[\r\nA = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) , \\ B = \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 2 \\end{array} \\right) , \\ E = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & 1 \\end{array} \\right)\r\n\\]\r\n\u306b\u3064\u3044\u3066, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(AB = BA\\) \u3092\u6e80\u305f\u3059 \\(A\\) \u306f, \\(x , y\\) \u3092\u7528\u3044\u3066 \\(A = x B +y E\\) \u3068\u8868\u305b\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(A^3 = E\\) \u306e\u3068\u304d\r\n\\[\r\n( t^2 -\\Delta ) A = ( t \\Delta +1 ) E\r\n\\]\r\n\u3092\u793a\u305b. \u305f\u3060\u3057, \\(t = a+d\\) , \\(\\Delta = ad -bc\\) \u3068\u3059\u308b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(AB = BA\\) \u304b\u3064 \\(A^3 = E\\) \u3092\u6e80\u305f\u3059 \\(A\\) \u3092\u3059\u3079\u3066\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n

    \u89e3\u7b54<\/h3>\r\n

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

    \\[\\begin{align}\r\nAB & = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 2 \\end{array} \\right) = \\left( \\begin{array}{cc} a-b & a+2b \\\\ c-d & c+2d \\end{array} \\right) , \\\\\r\nBA & = \\left( \\begin{array}{cc} 1 & 1 \\\\ -1 & 2 \\end{array} \\right) \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) = \\left( \\begin{array}{cc} a+c & b+d \\\\ -a+2c & -b+2d \\end{array} \\right)\r\n\\end{align}\\]\r\n\\(AB = BA\\) \u306a\u306e\u3067, \u5404\u6210\u5206\u3092\u6bd4\u8f03\u3059\u308b\u3068\r\n\\[\r\n\\left\\{ \\begin{array}{ll} a-b = a+c & ... [1] \\\\ a+2b = b+d & ... [2] \\\\ c-d = a+2c & ... [3] \\\\ c+2d = -b+2d & ... [4] \\end{array} \\right.\r\n\\]\r\n[1] [4] \u3088\u308a\r\n\\[\r\nc = -b\r\n\\]\r\n[2] [3] \u3088\u308a\r\n\\[\r\nd = a+b\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nA = \\left( \\begin{array}{cc} a & b \\\\ -b & a+b \\end{array} \\right) = b A +(a-b) E\r\n\\]\r\n

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

    \u30b1\u30fc\u30ea\u30fc\u30fb\u30cf\u30df\u30eb\u30c8\u30f3\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\r\nA^2 = t A -\\Delta E\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nA^3 & = \\left( t A -\\Delta E \\right) A \\\\\r\n& = t \\left( t A -\\Delta E \\right) -\\Delta A \\\\\r\n& = \\left( t^2 -\\Delta \\right) A -\\Delta t E = E\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\left( t^2 -\\Delta \\right) A = \\left( \\Delta t +1 \\right) E\r\n\\]\r\n

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

    \\[\\begin{align}\r\nt & = a +(a+b) = 2a +b , \\\\\r\n\\Delta & = a (a+b) +b^2 = a^2 +ab +b^2\r\n\\end{align}\\]\r\n(2)<\/strong> \u306e\u7d50\u679c\u304b\u3089, \u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(t^2 -\\Delta = 0\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nt^2 -\\Delta & = (2a+b)^2 -( a^2 +ab +b^2 ) \\\\\r\n& = 3a (a+b) = 0 \\\\\r\n\\text{\u2234} \\quad & a = 0 , \\ b = -a\r\n\\end{align}\\]\r\n