{"id":510,"date":"2012-12-07T23:41:19","date_gmt":"2012-12-07T14:41:19","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=510"},"modified":"2021-09-28T20:49:02","modified_gmt":"2021-09-28T11:49:02","slug":"thr200905","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr200905\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2009\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(a, b, c, d, p, q\\) \u306f \\(ad-bc \\gt 0 , \\ p \\gt 0 , \\ q \\gt 0\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b.\r\n\\(2\\) \u3064\u306e\u884c\u5217 \\(A = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right)\\) \u3068 \\(P = \\left( \\begin{array}{cc} p & 0 \\\\ 0 & q \\end{array} \\right)\\) \u304c \\(APA = P^2\\) \u3092\u6e80\u305f\u3059\u3068\u3059\u308b. \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\\(P^3 A = A P^3\\) \u304c\u6210\u308a\u7acb\u3064\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(A\\) \u3092 \\(p\\) \u3068 \\(q\\) \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

    \\[\r\nAPA = P^2 \\quad ... [1]\r\n\\]\r\n\\(ad-bc \\neq 0\\) \u3088\u308a \\(A^{-1}\\) \u306f\u5b58\u5728\u3059\u308b.
    \r\n[1] \u306e\u4e21\u8fba\u306b\u53f3\u304b\u3089 \\(A^{-1}\\) \u3092\u639b\u3051\u308b\u3068\r\n\\[\r\nAP = P^2 A^{-1} \\quad ... [2]\r\n\\]\r\n\u3088\u3063\u3066, [1] \u306e\u4e21\u8fba\u306b [2] \u306e\u5404\u8fba\u3092\u53f3\u304b\u3089\u639b\u3051\u308c\u3070\r\n\\[\\begin{align}\r\nP^2 A^{-1} APA & = AP P^2 \\\\\r\n\\text{\u2234} \\quad P^3 A & = A P^3\r\n\\end{align}\\]\r\n

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

    (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{cc} p^3 & 0 \\\\ 0 & q^3 \\end{array} \\right) \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) & = \\left( \\begin{array}{cc} a & b \\\\ c & d \\end{array} \\right) \\left( \\begin{array}{cc} p^3 & 0 \\\\ 0 & q^3 \\end{array} \\right) \\\\\r\n\\left( \\begin{array}{cc} a p^3 & b p^3 \\\\ c q^3 & d q^3 \\end{array} \\right) & = \\left( \\begin{array}{cc} a p^3 & b q^3 \\\\ c p^3 & d q^3 \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad b ( p^3 -q^3 ) = 0 , & \\ c ( p^3 -q^3 ) = 0\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u6e80\u305f\u3059\u306e\u306f\r\n\\[\r\np=q \\ \\text{\u307e\u305f\u306f} \\ b=c=0\r\n\\]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(p=q\\) \u306e\u3068\u304d
      \r\n\\(P = pE\\) \u3068\u8868\u305b\u3066, [1] \u306b\u4ee3\u5165\u3059\u308b\u3068\r\n\\[\\begin{align}\r\np A^2 & = p^2 E \\\\\r\n\\text{\u2234} \\quad A^2 -p E & = 0 \\quad ( \\ \\text{\u2235} \\ p \\neq 0 \\ )\r\n\\end{align}\\]\r\n\u30b1\u30fc\u30ea\u30fc\u30fb\u30cf\u30df\u30eb\u30c8\u30f3\u306e\u5b9a\u7406\u3088\u308a\r\n\\[\r\nad-bc = -p \\lt 0\r\n\\]\r\n\u3053\u308c\u306f\u6761\u4ef6\u306b\u77db\u76fe\u3059\u308b\u306e\u3067, \u4e0d\u9069\u3067\u3042\u308b.<\/p><\/li>\r\n

    2. 2*<\/strong>\u3000\\(b=c=0\\) \u306e\u3068\u304d
      \r\n\\(A = \\left( \\begin{array}{cc} a & 0 \\\\ 0 & d \\end{array} \\right)\\) \u3068\u8868\u305b\u3066, \\(ad \\gt 0 \\quad ... [3]\\) .\r\n\\[\r\nA^{-1} = \\dfrac{1}{ad} \\left( \\begin{array}{cc} d & 0 \\\\ 0 & a \\end{array} \\right) = \\left( \\begin{array}{cc} \\dfrac{1}{a} & 0 \\\\ 0 & \\dfrac{1}{d} \\end{array} \\right)\r\n\\]\r\n\u3053\u308c\u3089\u3092 [2] \u306b\u4ee3\u5165\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{cc} a & 0 \\\\ 0 & d \\end{array} \\right) \\left( \\begin{array}{cc} p & 0 \\\\ 0 & q \\end{array} \\right) & = \\left( \\begin{array}{cc} p^2 & 0 \\\\ 0 & q^2 \\end{array} \\right) \\left( \\begin{array}{cc} \\dfrac{1}{a} & 0 \\\\ 0 & \\dfrac{1}{d} \\end{array} \\right) \\\\\r\n\\left( \\begin{array}{cc} ap & 0 \\\\ 0 & dq \\end{array} \\right) & = \\left( \\begin{array}{cc} \\dfrac{p^2}{a} & 0 \\\\ 0 & \\dfrac{q^2}{d} \\end{array} \\right) \\\\\r\n\\text{\u2234} \\quad ap = \\dfrac{p^2}{a} , & \\ dq = \\dfrac{q^2}{d}\r\n\\end{align}\\]\r\n\\(p \\neq 0 , \\ q \\neq 0\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\na^2 = p , & \\ d^2 = q \\\\\r\n\\text{\u2234} \\quad a = \\pm \\sqrt{p} , & \\ d = \\pm \\sqrt{q}\r\n\\end{align}\\]\r\n\u305f\u3060\u3057, [3] \u3088\u308a \\(a\\) \u3068 \\(d\\) \u306f\u540c\u7b26\u53f7\u3067\u3042\u308b.<\/p><\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a\r\n\\[\r\nA = \\underline{\\left( \\begin{array}{cc} \\pm \\sqrt{p} & 0 \\\\ 0 & \\pm \\sqrt{q} \\end{array} \\right) \\quad ( \\text{\u8907\u53f7\u540c\u9806} )}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a, b, c, d, p, q\\) \u306f \\(ad-bc \\gt 0 , \\ p \\gt 0 , \\ q \\gt 0\\) \u3092\u6e80\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b. \\(2\\) \u3064\u306e\u884c\u5217 \\(A = \\left( \\begin{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":[75],"tags":[148,13],"class_list":["post-510","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2009","tag-tohoku_r","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/510","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=510"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/510\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=510"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=510"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=510"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}