{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"\n",
" 線形代数(Linear algebra) \n",
"
\n",
"\n",
" cc by Shigeto R. Nishitani, 2017-10-31 \n",
"
\n",
"* file: /Users/bob/python/doing_math_with_python/linear_algebra.ipynb"
]
},
{
"cell_type": "markdown",
"metadata": {
"toc": "true"
},
"source": [
"# Table of Contents\n",
" "
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# sympy\n",
"\n",
"* [sympy tutorial](http://docs.sympy.org/latest/tutorial/index.html)\n",
"* [module references](http://docs.sympy.org/latest/modules/matrices/matrices.html) LUdecompositionの記述あり．"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 行列，ベクトルの生成"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 直接的に"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZiK7\nRN38cqFhAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABPElEQVQ4Ee3Uv0vEMBQH8O/FK1TjtSf+2PT0\nFjdBEBzvwHPv5KqL52a7aNf6F+jkel3FRfwD5MBV0c1V/AcED11EY4QmfQ+amx3M0uZ9eHlpkgYt\n9YaKJpRqYra3XUHweltNzFXJb6xeaTK965a2HtHsHYgvY95BzGwfmDEGnDJbAuKurcdtELlN136N\nHHlAQ6+H+QY+JhDsuS3Vwzry/MRtm5BXjrx6At9lN+3VQ1vvuHWe6wqmDZT6tmaC7GnmyYJF599w\n3b7lK6N3otgjmaCTMwyGxvwRwmdmJ9Yai9g9oyYvrOkwH1NMEvNeaBouicm1I2peRgxYeSQowKw2\nT2yDmD6M06PSvGVi4TuziX4/XsiKdakNEXyWefpNRwqbytHJmIXW8JTeMxIPH7n9H5gUnT92Psfd\nkWPu1h85h1EamwQVaAAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 2\\\\3 & 4\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 2⎤\n",
"⎢ ⎥\n",
"⎣3 4⎦"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from sympy import *\n",
"\n",
"#init_session()\n",
"#init_printing(use_unicode=True)\n",
"init_printing()\n",
"list_a = [1,2]\n",
"list_b = [3,4]\n",
"A=Matrix([list_a, list_b])\n",
"A"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAyBAMAAAC0bgabAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhB2qyJEZt3N\nu5lcG7FTAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyElEQVQoFWOQ//+JAQz0/39kEHZxhXBCXAwZ\nRCBMEOlIHU5YemdSAtQ0VgWubdwHoBymAKavzA0wGQaeC0A7YZbyJyBx/B3gHOaAfAa2AKiy+Qb3\nGbqAUmCHhqeFvVoA44DUgwCVvAAxbABNY81LYGiEeZuJWYAhH+a5Zr4DDHdhnADGBIZvMA7D/AaW\nL3DOGwY2ATjnOgNvAZzzlmH+AjgnXO19A5zDwPAGyIZ4m/0A6084h3sBbwKcw5n2BMTGGzooSQw5\n8QEALm87XfZ6oCEAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}x\\\\y\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡x⎤\n",
"⎢ ⎥\n",
"⎣y⎦"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"x,y = symbols('x,y')\n",
"v1=Matrix([x,y])\n",
"v1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## ゼロ行列，単位行列"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAyBAMAAADfKoqLAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCZZs3dIrur\nRHbLQ9+lAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABHklEQVRIDWOQ//+JgQjA9P+/AIOwiysRShlY\nXZwFGESIUQlSwwJRG6augqIDjc+aVOQAV9vMELkAWTEan8OAdRZMLbsAA1MDklp0fiEDgwZMLfcB\nBuavSGrR+XsYGOwDoO7lP8DA+xlJLTr/DwPDfQOo2vgEBt5/SGrR+KzfgGofQNWuV2BgQY4QND4X\nUM4fqAQcZusT0NSi8rmA5sLVotnJgMZHcQPQL8xofkPh/wGGA8xv3BsY2FHCDI2/m4GhHhZmwLBn\na0AKB3Q+MC4yGKB+Y5jMEOiApBadz2nACkxh0LQT9O4IslIGND6r3hugUVC1KApxcUbVQkJmiIYD\nWtnBgMYfLUvAsTtalkCz/2hZglwOkpjnSWkTkNDWAACNI4zVcBPhcQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡0 0 0⎤\n",
"⎢ ⎥\n",
"⎣0 0 0⎦"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"zeros(2,3)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFUAAAAyBAMAAADfKoqLAAAAJFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMA74lUMhCrmXbNZj0HdO4AAAAJcEhZcwAA\nDsQAAA7EAZUrDhsAAACVSURBVEgNY5DevYmBCMC0e7cAg7CLKxFKGVhdnAUYRIhRCVLDQoZawwBU\n09H4zKJwc1nTs1HUovPDyrbC1TIwdKOoxeBzjKoFh/tgC4cy6WkLwA6DEmh89sSdWeSlM2RDcbHJ\nSb+4zEIWHzUXEhq0Dge0soMBjT9alkCiYbDleVqV1WhlBwMaf7QsQSujSGkTkNDWAAA4QlTBhcEl\ncwAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 1 & 1\\\\1 & 1 & 1\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 1 1⎤\n",
"⎢ ⎥\n",
"⎣1 1 1⎦"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ones(2,3)"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nxR99YOnjB5G2vSR/3mzMH1ja1v0A4UooEMbFTswAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 0 0⎤\n",
"⎢ ⎥\n",
"⎢0 1 0⎥\n",
"⎢ ⎥\n",
"⎣0 0 1⎦"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eye(3)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nopT4DxWtKn8oVmeOAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 2 & 0\\\\0 & 0 & 3\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 0 0⎤\n",
"⎢ ⎥\n",
"⎢0 2 0⎥\n",
"⎢ ⎥\n",
"⎣0 0 3⎦"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"diag(1,2,3)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 行列要素のとりだし，追加"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAAkAAAAOBAMAAAAPuiubAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAzXYQMplU74mrIma7\nRN0SDTw+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAATklEQVQIHWNgVGYJY2Bg//+/gYGBsVOLgYGB\nE4jh5G5VBwYGHgOuv2DBc2AyfgHDRAaG/gsMhQwM8QYMAQwMDxkYljJwVzEwcEVMNWAAADWrDnXF\n0pOGAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$5$$"
],
"text/plain": [
"5"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A=Matrix([[1,2,3],[4,5,6],[7,8,9]])\n",
"# i行j列の要素の取り出し\n",
"A[1,1]"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+WP1+ds3sqdW6BE639Zv3Nlc+klUxn4AAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}2 & 3\\\\5 & 6\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡2 3⎤\n",
"⎢ ⎥\n",
"⎣5 6⎦"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A[:2,1:4]"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRO/NZt27IolselnTNAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAQZJREFUOBHtlcFy\nxCAIhlHRbteYjeX937WiRscQZnrrJZwSP/lRVABD1SwM29sIgCGHxfwgkPjfEiMzR+dX+jvyMZx+\ni1ewztI9KvPxQSPzzkbabGpZXHJ4JvZBayb4TybKuD5Lov2jIYwaOtKmoQwaSoeGAoKGclncvaDn\n17mgr9d33Upgp4l+XvMte8tG0ZaIxWQ21C0DkLrlD1G/o1KwxrmP9aBeYM1uXWarGVlymFqhpla7\nF+SOOr1X9QU1odzlF1R93u347y7AecjybuRehqRXKC2h2zUWbicRguPhCcHQt8uuF8HSrzRBr6Mj\nzgZ3ERxqMtb/IrVlB27QiO+5vNqyEeEXlVQXSGZRLssAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}1\\\\4\\\\7\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1⎤\n",
"⎢ ⎥\n",
"⎢4⎥\n",
"⎢ ⎥\n",
"⎣7⎦"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"col_0 = A[:,0]\n",
"col_0"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFIAAAAZBAMAAABQnWQHAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iMpl272bNiRBU\nq0QwBS8SAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABFElEQVQ4EWMQUjJgIAz4lLQYFAgrA6uwJVWl\nK4rB0za4HkAW4Fq2hIEBaqYisgRD/f9fyHzueQyRMJVshcgyDLOuoXiSMYBBEKbSYiKKygsoPIb3\nYH0Q2xfgVSkO1ghWyWGAqvJGWwiyqT+9Uh9Abd/GgKpSmsEeKAUD3D8bGGqgKhegqWRgYFSAqWNg\n4P6/gSF8AziUeBwwVLJ+Rqhk+MTA0P8ArHIXA6pK3gkMXP+QVGoCVTaAVZ69e/f7NSQZ9gkMrL+R\n+NfhZgIFC5EkGFgZGJgmIAn4A90J9REDw0ckCQbmAIZ25EjiXcAN8zvDwv83kJWaro1G5jIsjXWA\nm4kigZ1DcvrEbgyK6NAxU4nI/K4LALy8QV1O9bHHAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\left[\\begin{matrix}4 & 5 & 6\\end{matrix}\\right]$$"
],
"text/plain": [
"[4 5 6]"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"row_1 = A[1,:]\n",
"row_1"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"numpyではcolumn_stackを使って拡大係数行列を作った．\n",
"sympyではrow_insert, col_insertを使う．"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABkCAMAAAA18wfQAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nCNFa/AMxQDWlW06BHAAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\\\1 & 2 & 3\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 2 3⎤\n",
"⎢ ⎥\n",
"⎢4 5 6⎥\n",
"⎢ ⎥\n",
"⎢7 8 9⎥\n",
"⎢ ⎥\n",
"⎣1 2 3⎦"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"b = Matrix([1,2,3])\n",
"A.row_insert(3,b.T)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAHgAAABLCAMAAACMVLPjAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nijUAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 2 & 3 & 1\\\\4 & 5 & 6 & 2\\\\7 & 8 & 9 & 3\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 2 3 1⎤\n",
"⎢ ⎥\n",
"⎢4 5 6 2⎥\n",
"⎢ ⎥\n",
"⎣7 8 9 3⎦"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.col_insert(3,b)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 行列・ベクトル間の単純な演算"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 次元の確認"
]
},
{
"cell_type": "code",
"execution_count": 46,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAD8AAAAUBAMAAADIGvgZAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAIma7zZnddlTvRIkQ\nqzLsm4+cAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABLElEQVQoFXWSMUvDUBDHf6/VmibVRkSXLtFP\noE7iYvwGAenmkEURHHRxNt+gg5PFIYuCW0FUpEsmxcmv4EdQEe0g6L2kgZfE/oe7d3e/5I57D5TH\nRC3rygJ0hhsVRg1vImZdyW+jAnpemVjE+qDlQy1gZkQ7LAOnEV9wDw0Xu8+LoEVdx61PeIalNF9t\ngW7BVMyJBlqHxc/T6HEATsC6BOputwp09hKZ0OcgLT3EqSsYpy9tQsSKmheFWhacJ8y9aUC51Edl\nYB62Ig1Ii/bPP8BvogE71EM2V2m8l/9wBkcuNZ9bGdWjd6zXYmoF6xumB1xJ9rK7CfuJWcfu7sSy\nqFe96kxWfjA52bNcVibHLORnuSzWxsFTnjS8vm79YFJ5Y2+6um6rPDETJE/uDw2rNiRV8o4HAAAA\nAElFTkSuQmCC\n",
"text/latex": [
"$$\\left ( 3, \\quad 3\\right )$$"
],
"text/plain": [
"(3, 3)"
]
},
"execution_count": 46,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.shape"
]
},
{
"cell_type": "code",
"execution_count": 49,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"(3, 1)\n",
"(1, 3)\n"
]
}
],
"source": [
"print(v1.shape)\n",
"print(v1.T.shape)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 転置(transpose)"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nYW0oDePnJwsAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 4 & 7\\\\2 & 5 & 8\\\\3 & 6 & 9\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 4 7⎤\n",
"⎢ ⎥\n",
"⎢2 5 8⎥\n",
"⎢ ⎥\n",
"⎣3 6 9⎦"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A.T"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAZBAMAAABjpnsOAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iEHarRIlmzVQy\nme/la43/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAu0lEQVQoFWMQUjJgwAaKlBoYFLBJgMSYE4aS\nHEtocmkryNlQwBIbwGDnwAz2gwfDpAAtmASQ5uB+wBDPAJFrYeh32IwkZ827gEEQKufAsBBJhoHB\ngS+A4StUjgHIQgH1Bjx/YHJAFgq4xcD6ACq3hfUDQwiypBgDYwJEjv034wfmCchyFxnqJ0DkWFJD\nYmciSzG4td43gMihCEM5t3DFH9sClu+45JgmMAbgkmMPvQM0mpk2aUIJZ/pMAACN0i4Uquj5/wAA\nAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}x & y\\end{matrix}\\right]$$"
],
"text/plain": [
"[x y]"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1.T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## ドット積(内積)"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFEAAAAyBAMAAADWwSrxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\ntQP/CfwGEG2XXvjNWNEAAAAASUVORK5CYII=\n",
"text/latex": [
"$$\\left[\\begin{matrix}x + 2 y\\\\3 x + 4 y\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ x + 2⋅y ⎤\n",
"⎢ ⎥\n",
"⎣3⋅x + 4⋅y⎦"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = Matrix([[1,2],[3,4]])\n",
"A*v1"
]
},
{
"cell_type": "code",
"execution_count": 29,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAaBAMAAADvbqfyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iEHarRIlmzVQy\nme/la43/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABS0lEQVQoFa3Tv0rDUBQG8K9JalLTahF1EyWg\n6JZJccsSV7MoLtWO4tRFcBDsCxR8hOAYl+6C9A3q4qCzLk6FFsSp5v45zYlcOpkh99zv/jg3uSQA\nXb2NiEo+6nglCGXqhQspX9a1iN1gF1t6boX1sYGpeL9gHfvbxGRcMMCfGJiKObMSIxMxZz2t/DIX\nMWO1FyOTMWM3uFWu3E3GksWnB2dJ/fFptcziV9gfKhbM6da2rXRpOh2VWfcZ1ZGKBfMib+KHioi7\n3tTuv6PSVLHshkU9yzMvyx6+sqwNOPjEclIwzGYqo1dwxxgOOKNZmTVSnKtEnpsfXaAa6SAfqFul\njXXG7vqbOCzUjDWa7g9jRyfx272BOZct+rTk8TIiStoUyJ9OXSbmhGrtCsedOUwvYQ9rVJq60dp1\na0DlPEYmH/+bBfo/ZVv8Ld1g5xcTZ0gndm5kQQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}x^{2} + y^{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ 2 2⎤\n",
"⎣x + y ⎦"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1.T*v1 #v1*v1ではerror"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ4AAAAZBAMAAAAyHoooAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nwZN8Gn/I5P/35ab6/zv3F3VIkZ00pX9FAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\left[\\begin{matrix}x + 3 y & 2 x + 4 y\\end{matrix}\\right]$$"
],
"text/plain": [
"[x + 3⋅y 2⋅x + 4⋅y]"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1.T*A"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAEwAAAAaBAMAAADvbqfyAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAu90iEHarRIlmzVQy\nme/la43/AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABS0lEQVQoFa3Tv0rDUBQG8K9JalLTahF1EyWg\n6JZJccsSV7MoLtWO4tRFcBDsCxR8hOAYl+6C9A3q4qCzLk6FFsSp5v45zYlcOpkh99zv/jg3uSQA\nXb2NiEo+6nglCGXqhQspX9a1iN1gF1t6boX1sYGpeL9gHfvbxGRcMMCfGJiKObMSIxMxZz2t/DIX\nMWO1FyOTMWM3uFWu3E3GksWnB2dJ/fFptcziV9gfKhbM6da2rXRpOh2VWfcZ1ZGKBfMib+KHioi7\n3tTuv6PSVLHshkU9yzMvyx6+sqwNOPjEclIwzGYqo1dwxxgOOKNZmTVSnKtEnpsfXaAa6SAfqFul\njXXG7vqbOCzUjDWa7g9jRyfx272BOZct+rTk8TIiStoUyJ9OXSbmhGrtCsedOUwvYQ9rVJq60dp1\na0DlPEYmH/+bBfo/ZVv8Ld1g5xcTZ0gndm5kQQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}x^{2} + y^{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ 2 2⎤\n",
"⎣x + y ⎦"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1.T*v1"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAE4AAAAzBAMAAADLOCMxAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nhLMgV5TnzL/qtiCa/6vmnyZ5N2T/9F8HE5aKuGYB1AAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}x^{2} & x y\\\\x y & y^{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡ 2 ⎤\n",
"⎢x x⋅y⎥\n",
"⎢ ⎥\n",
"⎢ 2 ⎥\n",
"⎣x⋅y y ⎦"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1*v1.T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 外積，outer, cross\n",
"\n",
"outer(v1,v2)がSympy1.1.1では見つからない．\n",
"crossは使える．"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": [
"v1 = Matrix([1,1,3])\n",
"v2 = Matrix([1,2,-1])"
]
},
{
"cell_type": "code",
"execution_count": 31,
"metadata": {},
"outputs": [
{
"ename": "AttributeError",
"evalue": "MutableDenseMatrix has no attribute outer.",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mAttributeError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mv1\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mouter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mv2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m/Users/bob/anaconda3/lib/python3.6/site-packages/sympy/matrices/matrices.py\u001b[0m in \u001b[0;36m__getattr__\u001b[0;34m(self, attr)\u001b[0m\n\u001b[1;32m 3214\u001b[0m \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3215\u001b[0m raise AttributeError(\n\u001b[0;32m-> 3216\u001b[0;31m \"%s has no attribute %s.\" % (self.__class__.__name__, attr))\n\u001b[0m\u001b[1;32m 3217\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 3218\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0mintegrate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mAttributeError\u001b[0m: MutableDenseMatrix has no attribute outer."
]
}
],
"source": [
"v1.outer(v2)"
]
},
{
"cell_type": "code",
"execution_count": 32,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAACoAAABLCAMAAAAxteWWAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRM3diWYiu+9sJ/xb8wAAAAlwSFlzAAAOxAAADsQBlSsOGwAAASNJREFUSA3tl9uS\ngyAMQMO13aJYNv//r8vFCFML5LVd86BxOBNyIw4gMIuEriyFABCodBTTJcGndYkJFX2qrvgBKhap\n1iQFH6G+eIh7ECNUbdkchTtCy8YrBTJCs82HpsCm6JNImKGr56I25pKktWrdcogrwWhHIMwcWKqr\nE9RS+pPt1oG6F2mxkUidoYaPbqHpzLEDdfeoXeh3ZEA0TZUL3A+r7eoxqkNzAIbo5h0XXYGL+o2L\n2jh9mFbTYOOhJs2Lt+jt/pOzQQ+bp+UJ/b2f/wVGJsEgj3mZjfSrxS4BALJL8ESUdbYmD/oOUJDH\n+0L/WQZMsEfxd+VtBqxUEnloNKMv9GMyoGRA9zIxrpGxZ4B9hbDpgqD14/Uw1e98hdAa/gCTGhgW\ntC2zCQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}-7\\\\4\\\\1\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡-7⎤\n",
"⎢ ⎥\n",
"⎢4 ⎥\n",
"⎢ ⎥\n",
"⎣1 ⎦"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1.cross(v2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## スカラー三重積"
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAVO8Qq5l2zWaJMkS7\nIt2ZnNffAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAcElEQVQIHWNgAAPOabOANKMyAwNzNUM4A4NJ\nyGcGBp4AhjNAUTYgc/8FsDIQ8xCYBRb9vjJ1A0QB8/cGBkUo878BQ5ABRNs/BoZ+oAqQtudAZgOE\n6YQQXQ9UC9HGwDKBGWgCq9N3DwaGqSELgKJwAADSWSGR+xZb5QAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$16$$"
],
"text/plain": [
"16"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v3 = Matrix([-1,2,1])\n",
"v3.dot(v1.cross(v2))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 行列操作"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 掃き出し, LU分解"
]
},
{
"cell_type": "code",
"execution_count": 42,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"[Matrix([\n",
"[ 1],\n",
"[-2],\n",
"[ 1]])]\n",
"⎡1 2 3 1⎤\n",
"⎢ ⎥\n",
"⎢4 5 6 2⎥\n",
"⎢ ⎥\n",
"⎣5 7 9 3⎦\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAASAAAABNCAMAAAAxQEmCAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nSf2jye72OMc32jViWxKF/gv++lE2JKNo8AAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left ( \\left[\\begin{matrix}1 & 0 & -1 & - \\frac{1}{3}\\\\0 & 1 & 2 & \\frac{2}{3}\\\\0 & 0 & 0 & 0\\end{matrix}\\right], \\quad \\left [ 0, \\quad 1\\right ]\\right )$$"
],
"text/plain": [
"⎛⎡1 0 -1 -1/3⎤, [0, 1]⎞\n",
"⎜⎢ ⎥ ⎟\n",
"⎜⎢0 1 2 2/3 ⎥ ⎟\n",
"⎜⎢ ⎥ ⎟\n",
"⎝⎣0 0 0 0 ⎦ ⎠"
]
},
"execution_count": 42,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A=Matrix([[1,2,3],[4,5,6],[5,7,9]])\n",
"print(A.nullspace())\n",
"Aex = A.col_insert(3,b)\n",
"pprint(Aex)\n",
"Aex.rref()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"rrefはreduced row echelon formを取り出してくれる．これはGaussの消去法で下んとこまで行って，さらに上んとこへ戻ってきたやつ．拡張行列もそのままやってくれるはず．もうすこし，試す必要があるが，一応MapleのLUDecompositionに近い結果を与えてくれる．"
]
},
{
"cell_type": "code",
"execution_count": 43,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAVQAAABLCAMAAAAYu499AAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nTkSuQmCC\n",
"text/latex": [
"$$\\left ( \\left[\\begin{matrix}1 & 0 & 0\\\\4 & 1 & 0\\\\7 & \\frac{5}{3} & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1 & 2 & 3\\\\0 & -3 & -6\\\\0 & 0 & -3\\end{matrix}\\right], \\quad \\left [ \\right ]\\right )$$"
],
"text/plain": [
"⎛⎡1 0 0⎤, ⎡1 2 3 ⎤, []⎞\n",
"⎜⎢ ⎥ ⎢ ⎥ ⎟\n",
"⎜⎢4 1 0⎥ ⎢0 -3 -6⎥ ⎟\n",
"⎜⎢ ⎥ ⎢ ⎥ ⎟\n",
"⎝⎣7 5/3 1⎦ ⎣0 0 -3⎦ ⎠"
]
},
"execution_count": 43,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A=Matrix([[1,2,3],[4,5,6],[7,9,8]])\n",
"L, U, P = A.LUdecomposition()\n",
"L, U, P"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 階数, rank"
]
},
{
"cell_type": "code",
"execution_count": 26,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Matrix([[1, 2, 3], [4, 5, 6], [7, 9, 8]])\n",
"3\n"
]
}
],
"source": [
"print(A)\n",
"\n",
"print(A.rank())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 行列式, determinant"
]
},
{
"cell_type": "code",
"execution_count": 27,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"1\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABkAAAAOBAMAAAAoFKpzAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAEM3dMiKZu6uJRO92VKoV\n2MsAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABeSURBVAgdY2CAAta0cgcYm4FBgoHzFYLXyMCwG8Hb\nzMDQh1CqJ4DMA6qyE0AoZeB6icRhYG5gYBAyBgEVoHAashRPAjLvGgNrAJzPksDAg+AFz5zWDJdi\n0Hv37jEDAGBWEJ9xq8kjAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$-2$$"
],
"text/plain": [
"-2"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c = Matrix([[1,1],[1,2]])\n",
"print(c.det())\n",
"\n",
"a = Matrix([[1,2],[3,4]])\n",
"a.det()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 課題"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"次の行列式の計算を確認せよ．\n",
"$$ \n",
"\\left|\n",
"\\begin {array}{cccc} \n",
"1+a & 1 & 1 & 1\\\\\n",
"1 & 1+b & 1 & 1\\\\\n",
"1 & 1 & 1+c & 1\\\\\n",
"1 & 1 & 1 & 1+d\n",
"\\end {array} \\right|\n",
"= abcd \\left(1+\\sum_{i=a}^d \\frac{1}{i}\\right)\n",
"$$\n"
]
},
{
"cell_type": "code",
"execution_count": 59,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"⎡a + 1 1 1 1 ⎤\n",
"⎢ ⎥\n",
"⎢ 1 b + 1 1 1 ⎥\n",
"⎢ ⎥\n",
"⎢ 1 1 c + 1 1 ⎥\n",
"⎢ ⎥\n",
"⎣ 1 1 1 d + 1⎦\n"
]
},
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAQQAAAAQBAMAAAACFhwAAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n6L+ZYANVkKq6NgaNkznYPxoXEiWn1O6VAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$a b c d + a b c + a b d + a c d + b c d$$"
],
"text/plain": [
"a⋅b⋅c⋅d + a⋅b⋅c + a⋅b⋅d + a⋅c⋅d + b⋅c⋅d"
]
},
"execution_count": 59,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from sympy import *\n",
"a,b,c,d,x,y,z = symbols('a b c d x y z')\n",
"\n",
"A = Matrix([[1+a,1,1,1],[1,1+b,1,1],[1,1,1+c,1],[1,1,1,1+d]]\n",
")\n",
"pprint(A)\n",
"simplify(A.det())"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## 逆行列"
]
},
{
"cell_type": "code",
"execution_count": 60,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFkAAAAyBAMAAADFFmoFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n+exeR7MvjpTvMhG/FKGXifvF0dXXzAcqTG6s+l48eQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}-2 & 1\\\\\\frac{3}{2} & - \\frac{1}{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡-2 1 ⎤\n",
"⎢ ⎥\n",
"⎣3/2 -1/2⎦"
]
},
"execution_count": 60,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = Matrix([[1,2],[3,4]])\n",
"A.inv()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 課題\n",
"次の連立方程式を解け．\n",
"$$ \n",
"\\left\\{\n",
"\\begin {array}{cl} \n",
"x+y+z&=0\\\\\n",
"ax+by+cz&=0\\\\\n",
"bcx+cay+abz&=(a-b)(b-c)(c-a)\n",
"\\end {array} \\right.\n",
"$$"
]
},
{
"cell_type": "code",
"execution_count": 28,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"Matrix([[1, 1, 1], [a, b, c], [b*c, a*c, a*b]])\n",
"Matrix([[0], [0], [(-a + c)*(a - b)*(b - c)]])\n",
"-b + c\n",
"a - c\n",
"-a + b\n"
]
}
],
"source": [
"from pprint import pprint\n",
"from sympy import *\n",
"a,b,c,x,y,z = symbols('a b c x y z')\n",
"\n",
"A = Matrix([[1,1,1],[a,b,c],[b*c,c*a,a*b]])\n",
"bb = Matrix([0,0,(a-b)*(b-c)*(c-a)])\n",
"print(A)\n",
"print(bb)\n",
"Ainv = A.inv()\n",
"res = Ainv * bb\n",
"pprint(simplify(res[0]))\n",
"pprint(simplify(res[1]))\n",
"pprint(simplify(res[2]))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# 固有値，固有ベクトル\n",
"\n",
"sympyでは固有値・固有ベクトルは，eigenvects()で求められます．\n",
"戻り値は，\n",
"> [(固有値, 重複度, [固有ベクトル]), ...]\n",
"\n",
"です．"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAZkAAAAyBAMAAAB8J7tkAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left [ \\left ( -1, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}-1\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 3, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}1\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
],
"text/plain": [
"⎡⎛-1, 1, ⎡⎡-1⎤⎤⎞, ⎛3, 1, ⎡⎡1⎤⎤⎞⎤\n",
"⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥⎥⎟⎥\n",
"⎣⎝ ⎣⎣1 ⎦⎦⎠ ⎝ ⎣⎣1⎦⎦⎠⎦"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = Matrix([[1,2],[2,1]])\n",
"A.eigenvects()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## ベクトルの規格化\n",
"\n",
"eigenvectsの固有ベクトルは規格化されていません．norm()を使うとかんたんに規格化してくれます．"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABQAAAAZBAMAAADDORPgAAAALVBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADAOrOgAAAADnRSTlMAu90imTIQq4lE7812VGR7\nQmkAAAAJcEhZcwAADsQAAA7EAZUrDhsAAABwSURBVBgZYxBSMmAAAk4lDQYFEAMEzNGYzC6CAVDR\ndAaOl1BmGQPDKihzEQNDXQBE274EOBNokl4CzDDOF1C1DAxMBXCmC9w2Hgc48woDswFEG6sDAw+U\nadTRUgTVtu/du0dwE4C6YFZQxFSChYM6AIqaGQAdlbreAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\left[\\begin{matrix}2\\end{matrix}\\right]$$"
],
"text/plain": [
"[2]"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1=Matrix(A.eigenvects()[0][2])\n",
"v2=Matrix(A.eigenvects()[1][2])\n",
"\n",
"simplify(v1.T*v1)"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAABsAAAAVBAMAAABF8IgWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAInarRM2ZVBDdiWbv\nuzJCz3LGAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAtElEQVQYGWNggADG/yDwAcpjYHaAscC0KAqP\nIR2FyzEBhctWgMLtZGAQy70HF1rNwGHAsN4BygdqZW5g4DkA5TJtYOABivwDcl2AmJOBgfUHmMt0\nFMhVAqvi/MHAIKHvwMAAsdV/AVC0/wIDN4hmYJgMItg+MnA2gBkJIJLxG8MuEM2gBiYZ7BtmgBjs\nCQyCIPr98gQQ9YyBQRFEsx5tAJKMs1Ylgy1g/wYWBIYNxL5AEBcCALUeKW//iOF2AAAAAElFTkSu\nQmCC\n",
"text/latex": [
"$$\\sqrt{2}$$"
],
"text/plain": [
"√2"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v_norm = v1.norm()\n",
"v_norm"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADkAAAA/BAMAAACob32rAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nfi2U3/O/b8lT+y1ZNX/qQImtliOUkwAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}- \\frac{\\sqrt{2}}{2}\\\\\\frac{\\sqrt{2}}{2}\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡-√2 ⎤\n",
"⎢────⎥\n",
"⎢ 2 ⎥\n",
"⎢ ⎥\n",
"⎢ √2 ⎥\n",
"⎢ ── ⎥\n",
"⎣ 2 ⎦"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1n = v1/v_norm\n",
"v1n"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": true
},
"source": [
"## 対角化\n",
"\n",
"ちょいとわかりやすい行列を作成してみてみます．"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAARcAAAAyBAMAAACE64LWAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n7v/84if6C/3w9eq86LO3AAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\left ( \\left[\\begin{matrix}1 & 1\\\\0 & 2\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1 & 1\\\\0 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1 & 0\\\\0 & 2\\end{matrix}\\right]\\right )$$"
],
"text/plain": [
"⎛⎡1 1⎤, ⎡1 1⎤, ⎡1 0⎤⎞\n",
"⎜⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎟\n",
"⎝⎣0 2⎦ ⎣0 1⎦ ⎣0 2⎦⎠"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"D = diag(1,2)\n",
"P = Matrix([[1,0],[1,1]]).T\n",
"A = P*D*P.inv()\n",
"A, P, D"
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 0\\\\0 & 2\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 0⎤\n",
"⎢ ⎥\n",
"⎣0 2⎦"
]
},
"execution_count": 20,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P,D = A.diagonalize()\n",
"\n",
"P.inv()*A*P\n"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZt0i\nu0TKQvJ9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA00lEQVQ4EWOQ//+JAQtg+v9fgEHYxRWLFAOr\ni7MAgwg2GZAYCxFyhgEouplF4fpY0/NR5MLKvsLlGBj6UeQYOEblwPFQJj9tAVKAsif+zCIuHpA0\nwZnExB9cMRJjGOkL11JB8hiUCfXfZIboA8iSSGmXS4CBaQKSHHLa5U1gYAalVjhASrv8CQx8n+ES\nQAaSXHwBA98PHHLnFRhYUDIvkr7zBbjl8JkJdAszLrfwXmDgwuUHoN/Zkf2O7AeGzQyBDjj8wBDU\nkYIsNeTSLr4yEk/ZCgD8YE0m/VSorQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 1\\\\0 & 1\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 1⎤\n",
"⎢ ⎥\n",
"⎣0 1⎦"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"eigs = A.eigenvects()\n",
"v1 = eigs[0][-1][0]\n",
"v2 = eigs[1][-1][0]\n",
"\n",
"Matrix([v1,v2]).reshape(2,2).T"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"対角化行列$P$をeigenvectsから作ろうとして苦労した．まず，固有ベクトルの取り出しが自動的にできない．重複しているときには，固有ベクトルは二つ出てくる．さらに，タプルからの取り出しがこれでいいいのか．．．さらに，ベクトルから行列を作るときに横置きのができない．うううんん．他のやり方をしたでやってみる．上では仕方がないので，一旦4x1のMatrixにしてそのあとreshape，さらにTransposeで．．．"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUMhCrmXbNZt0i\nu0TKQvJ9AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA00lEQVQ4EWOQ//+JAQtg+v9fgEHYxRWLFAOr\ni7MAgwg2GZAYCxFyhgEouplF4fpY0/NR5MLKvsLlGBj6UeQYOEblwPFQJj9tAVKAsif+zCIuHpA0\nwZnExB9cMRJjGOkL11JB8hiUCfXfZIboA8iSSGmXS4CBaQKSHHLa5U1gYAalVjhASrv8CQx8n+ES\nQAaSXHwBA98PHHLnFRhYUDIvkr7zBbjl8JkJdAszLrfwXmDgwuUHoN/Zkf2O7AeGzQyBDjj8wBDU\nkYIsNeTSLr4yEk/ZCgD8YE0m/VSorQAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 1\\\\0 & 1\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 1⎤\n",
"⎢ ⎥\n",
"⎣0 1⎦"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v1.col_insert(1,v2)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"numpyにはravel()というフラット化の関数があるがsympyにはない．\n",
"\n",
"結論：3つほど上でやっているdiagonalizeで取り出しましょう．重複があっても取り出してくれるんで．"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### 課題\n",
"行列\n",
"$$\n",
"A\\, := \\, \\left[\n",
"\\begin {array}{ccc} \n",
"2&0&1\\\\\n",
"0&3&0\\\\\n",
"1&0&2\n",
"\\end {array} \\right]\n",
"$$\n",
"\n",
"を対角化する変換行列$P$を求め，対角化せよ．"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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LB/ZJfYQaOinFkm601TykIarzE+3/A0LuvFGLjj9rAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\left [ \\left ( 1, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}-1\\\\0\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 3, \\quad 2, \\quad \\left [ \\left[\\begin{matrix}0\\\\1\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\0\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
],
"text/plain": [
"⎡⎛1, 1, ⎡⎡-1⎤⎤⎞, ⎛3, 2, ⎡⎡0⎤, ⎡1⎤⎤⎞⎤\n",
"⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥ ⎢ ⎥⎥⎟⎥\n",
"⎢⎜ ⎢⎢0 ⎥⎥⎟ ⎜ ⎢⎢1⎥ ⎢0⎥⎥⎟⎥\n",
"⎢⎜ ⎢⎢ ⎥⎥⎟ ⎜ ⎢⎢ ⎥ ⎢ ⎥⎥⎟⎥\n",
"⎣⎝ ⎣⎣1 ⎦⎦⎠ ⎝ ⎣⎣0⎦ ⎣1⎦⎦⎠⎦"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"A = Matrix([[2,0,1],[0,3,0],[1,0,2]])\n",
"evs = A.eigenvects()\n",
"evs\n",
"#"
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAARQAAABLCAMAAACGAYjdAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n/wOYX2slujVXlwAAAABJRU5ErkJggg==\n",
"text/latex": [
"$$\\left ( \\left[\\begin{matrix}-1 & 0 & 1\\\\0 & 1 & 0\\\\1 & 0 & 1\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1 & 0 & 0\\\\0 & 3 & 0\\\\0 & 0 & 3\\end{matrix}\\right]\\right )$$"
],
"text/plain": [
"⎛⎡-1 0 1⎤, ⎡1 0 0⎤⎞\n",
"⎜⎢ ⎥ ⎢ ⎥⎟\n",
"⎜⎢0 1 0⎥ ⎢0 3 0⎥⎟\n",
"⎜⎢ ⎥ ⎢ ⎥⎟\n",
"⎝⎣1 0 1⎦ ⎣0 0 3⎦⎠"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P,D = A.diagonalize()\n",
"P,D"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQfrA0rbmCy2KKwb1Df0TAAAAAElFTkSuQmCC\n",
"text/latex": [
"$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 3 & 0\\\\0 & 0 & 3\\end{matrix}\\right]$$"
],
"text/plain": [
"⎡1 0 0⎤\n",
"⎢ ⎥\n",
"⎢0 3 0⎥\n",
"⎢ ⎥\n",
"⎣0 0 3⎦"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"P.inv()*A*P"
]
},
{
"cell_type": "code",
"execution_count": null,
"metadata": {
"collapsed": true
},
"outputs": [],
"source": []
}
],
"metadata": {
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.6.1"
},
"latex_envs": {
"LaTeX_envs_menu_present": true,
"autocomplete": true,
"bibliofile": "biblio.bib",
"cite_by": "apalike",
"current_citInitial": 1,
"eqLabelWithNumbers": true,
"eqNumInitial": 1,
"hotkeys": {
"equation": "Ctrl-E",
"itemize": "Ctrl-I"
},
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},
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},
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},
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},
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}
},
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}