{"id":1957,"date":"2019-06-24T18:13:59","date_gmt":"2019-06-24T09:13:59","guid":{"rendered":"https:\/\/www.bi.appi.keio.ac.jp\/?p=1957"},"modified":"2024-08-20T18:16:32","modified_gmt":"2024-08-20T09:16:32","slug":"%e5%82%be%e3%81%84%e3%81%9f%e6%a5%95%e5%86%86%e3%81%ae%e5%bc%8f","status":"publish","type":"post","link":"https:\/\/www.bi.appi.keio.ac.jp\/?p=1957","title":{"rendered":"\u50be\u3044\u305f\u6955\u5186\u306e\u5f0f"},"content":{"rendered":"\n

\u7269\u7406\u60c5\u5831\u6570\u5b66B\u3067\u5bfe\u79f0\u884c\u5217\u306e\u5fdc\u7528\u4f8b\u3068\u3057\u3066\u53d6\u308a\u4e0a\u3052\u3066\u3044\u308b\u50be\u3044\u305f\u6955\u5186\u306e\u5f0f\u306e\u8aac\u660e\u3067\u3059\uff0e<\/p>\n\n\n\n

$5x^2 + 8xy + 5y^2 = 1$
\u306e\u5f0f\u3092\u884c\u5217\u3067\u8868\u3059\u3068
$\\left[\\begin{array}{cc}x & y\\end{array}\\right]\\left[\\begin{array}{cc}5 & 4\\\\ 4 & 5\\end{array}\\right]\\left[\\begin{array}{c}x\\\\ y\\end{array}\\right]=1$
\u306b\u306a\u308b\uff0e<\/p>\n\n\n\n

\u884c\u5217\u3092$A$\u3068\u7f6e\u304f\u3068\uff0c$A$\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\uff08\u6b63\u898f\u5316\u3055\u308c\u305f\u3082\u306e\uff09\u3092\u5217\u306b\u6301\u3064\u884c\u5217$Q$\u3092\u7528\u3044\u3066$A=Q\\Lambda Q^T$\u3068\u8868\u305b\u308b\uff0e\u884c\u5217$A$\u306f\u56fa\u6709\u5024\u3068\u3057\u30669\u30681\u3092\u6301\u3064\uff0e\u56fa\u6709\u5024\u304c9\u306e\u3068\u304d\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306f$[1\\ 1]^T$\u3067\u3042\u308b\uff0e\u56fa\u6709\u5024\u304c1\u306e\u3068\u304d\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306f$[1\\ -1]^T$\u3067\u3042\u308b\uff0e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u306e\u5927\u304d\u3055\u306f$\\sqrt{2}$\u3067\u3042\u308b\u306e\u3067
$\\left[\\begin{array}{cc}x & y\\end{array}\\right]\\displaystyle\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{rr}1 & 1\\\\ 1 & -1\\end{array}\\right]\\left[\\begin{array}{cc}9 & 0\\\\ 0 & 1\\end{array}\\right]\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{rr}1 & 1\\\\ 1 & -1\\end{array}\\right]\\left[\\begin{array}{c}x\\\\ y\\end{array}\\right]=1$
\u3068\u306a\u308b\uff0e<\/p>\n\n\n\n

\u3053\u306e\u5f0f\u306f
$9\\left(\\displaystyle\\frac{x+y}{\\sqrt{2}}\\right)^2 + 1\\left(\\displaystyle\\frac{x-y}{\\sqrt{2}}\\right)^2=1$
\u3068\u8868\u305b\u308b\uff0e\u62ec\u5f27\u5185\u3092$X$\u3068$Y$\u3068\u7f6e\u304f\u3068\uff0c
$9X^2+Y^2=1$
\u306e\u6955\u5186\u306e\u5f0f\u306b\u306a\u308b\uff0e\u6955\u5186\u306e\u77ed\u534a\u5f84\u306f1\/3\u3067\u9577\u534a\u5f84\u306f1\u306b\u306a\u308b\uff0e\u3053\u308c\u3089\u306f\uff0c\u56fa\u6709\u5024\u306e\u9006\u6570\u306e\u5e73\u65b9\u6839\u3067\u3042\u308b\uff0e<\/p>\n\n\n\n

import numpy as np\nimport matplotlib.pyplot as plt\nimport numpy.linalg as LA\n\n#\n# 5x^2 + 8xy + 5y^2 = 1\n# 9X^2 + y^2 = 1\n# X = (x + y)\/sqrt(2)\n# Y = (x - y) \/sqrt(2)\n#\ntheta = np.arange(0, 361, 3)\nX = np.cos(theta\/180*np.pi)\/3\nY = np.sin(theta\/180*np.pi)\nx = (X+Y)\/np.sqrt(2)\ny = (X-Y)\/np.sqrt(2)\nplt.figure(figsize=(5, 5))\nplt.xlim([-1.2, 1.2])\nplt.ylim([-1.2, 1.2])\nplt.xlabel('x')\nplt.ylabel('y')\nplt.plot(x, y, color='blue', alpha=0.8)\n#\nA = np.array([[5, 4], [4, 5]])\nw,v = LA.eig(A)\nprint('\u56fa\u6709\u5024')\nprint(w)\nprint('\u56fa\u6709\u30d9\u30af\u30c8\u30eb')\nprint(v)\n# \u56fa\u6709\u5024\u306e\u9006\u6570\u306e\u5e73\u65b9\u6839\u304c\u9577\u534a\u5f84\u3068\u77ed\u534a\u5f84\n# \u56fa\u6709\u30d9\u30af\u30c8\u30eb\u304c\u8ef8\u65b9\u5411\na = v[:,0].reshape(2, 1) \/ np.sqrt(w[0])\nb = v[:,1].reshape(2 ,1) \/ np.sqrt(w[1])\nplt.plot(np.array([0, a[0]]), np.array([0, a[1]]), color='red')\nplt.plot(np.array([0, b[0]]), np.array([0, b[1]]), color='green')\nplt.show()\n<\/pre>\n\n\n\n
\u56fa\u6709\u5024\n[9. 1.]\n\u56fa\u6709\u30d9\u30af\u30c8\u30eb\n[[ 0.70710678 -0.70710678]\n [ 0.70710678  0.70710678]]<\/pre>\n\n\n\n
\"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"
\u7269\u7406\u60c5\u5831\u6570\u5b66B\u3067\u5bfe\u79f0\u884c\u5217\u306e\u5fdc\u7528\u4f8b\u3068\u3057\u3066\u53d6\u308a\u4e0a\u3052\u3066\u3044\u308b\u50be\u3044\u305f\u6955\u5186\u306e\u5f0f\u306e\u8aac\u660e\u3067\u3059\uff0e $5x^2 + 8xy + 5y^2 = 1$\u306e\u5f0f\u3092\u884c\u5217\u3067\u8868\u3059\u3068$\\left[\\begin{array}{cc}x & y\\end{ … “\u50be\u3044\u305f\u6955\u5186\u306e\u5f0f” \u306e<\/span>\u7d9a\u304d\u3092\u8aad\u3080<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-1957","post","type-post","status-publish","format-standard","hentry","category-9"],"_links":{"self":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/1957","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1957"}],"version-history":[{"count":27,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/1957\/revisions"}],"predecessor-version":[{"id":2252,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=\/wp\/v2\/posts\/1957\/revisions\/2252"}],"wp:attachment":[{"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1957"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=1957"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.bi.appi.keio.ac.jp\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=1957"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}