{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### eigenvectsに関するコメント\n",
    "\n",
    "最後の固有値を求めるeigenvectsで失敗するかもしれません．\n",
    "これはsympyのversionによるようです．1.6.1ではダメです．\n",
    "それまでのは行けそうです．\n",
    "\n",
    "上の方で，対角ベクトルを与えるところで，\n",
    "Rationalを使うとちゃんと求めてくれます．\n",
    "```python\n",
    "#V = [1/5,1,1/2,1/3,1/4,1/2,1]\n",
    "V = [Rational(1,5),1,Rational(1,2),Rational(1,3),Rational(1,4),Rational(1,2),1]\n",
    "```\n",
    "\n",
    "これに関する問い合わせと開発者の反応は[Githubのissue](https://github.com/sympy/sympy/issues/20351) の通りです．\n",
    "この辺りがOSS開発ソフトで時々おこる問題です．\n",
    "そういうものだということを覚悟して，\n",
    "いくつかのライブラリや他のソフトで結果をクロス検証することを心がけてください．\n",
    "\n",
    "<2020-11-02 月>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1 & 1 & 1 & 1 & 0 & 1\\\\1 & 0 & 0 & 0 & 0 & 0 & 0\\\\1 & 1 & 0 & 0 & 0 & 0 & 0\\\\0 & 1 & 1 & 0 & 1 & 0 & 0\\\\1 & 0 & 1 & 1 & 0 & 1 & 0\\\\1 & 0 & 0 & 0 & 1 & 0 & 0\\\\0 & 0 & 0 & 0 & 1 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0  1  1  1  1  0  1⎤\n",
       "⎢                   ⎥\n",
       "⎢1  0  0  0  0  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎢1  1  0  0  0  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎢0  1  1  0  1  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎢1  0  1  1  0  1  0⎥\n",
       "⎢                   ⎥\n",
       "⎢1  0  0  0  1  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎣0  0  0  0  1  0  0⎦"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "M = Matrix([[0, 1, 1, 1, 1, 0, 1],\n",
    "[1, 0, 0, 0, 0, 0, 0],\n",
    "[1, 1, 0, 0, 0, 0, 0],\n",
    "[0, 1, 1, 0, 1, 0, 0],\n",
    "[1, 0, 1, 1, 0, 1, 0],\n",
    "[1, 0, 0, 0, 1, 0, 0],\n",
    "[0, 0, 0, 0, 1, 0, 0]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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BWZjjBQAAAABJRU5ErkJggg==",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1 & 1 & 0 & 1 & 1 & 0\\\\1 & 0 & 1 & 1 & 0 & 0 & 0\\\\1 & 0 & 0 & 1 & 1 & 0 & 0\\\\1 & 0 & 0 & 0 & 1 & 0 & 0\\\\1 & 0 & 0 & 1 & 0 & 1 & 1\\\\0 & 0 & 0 & 0 & 1 & 0 & 0\\\\1 & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡0  1  1  0  1  1  0⎤\n",
       "⎢                   ⎥\n",
       "⎢1  0  1  1  0  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎢1  0  0  1  1  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎢1  0  0  0  1  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎢1  0  0  1  0  1  1⎥\n",
       "⎢                   ⎥\n",
       "⎢0  0  0  0  1  0  0⎥\n",
       "⎢                   ⎥\n",
       "⎣1  0  0  0  0  0  0⎦"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Mt = M.T\n",
    "Mt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}\\frac{1}{5} & 0 & 0 & 0 & 0 & 0 & 0\\\\0 & 1 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & \\frac{1}{2} & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & \\frac{1}{3} & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & \\frac{1}{4} & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & \\frac{1}{2} & 0\\\\0 & 0 & 0 & 0 & 0 & 0 & 1\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡1/5  0   0    0    0    0   0⎤\n",
       "⎢                             ⎥\n",
       "⎢ 0   1   0    0    0    0   0⎥\n",
       "⎢                             ⎥\n",
       "⎢ 0   0  1/2   0    0    0   0⎥\n",
       "⎢                             ⎥\n",
       "⎢ 0   0   0   1/3   0    0   0⎥\n",
       "⎢                             ⎥\n",
       "⎢ 0   0   0    0   1/4   0   0⎥\n",
       "⎢                             ⎥\n",
       "⎢ 0   0   0    0    0   1/2  0⎥\n",
       "⎢                             ⎥\n",
       "⎣ 0   0   0    0    0    0   1⎦"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "#V = [1/5,1,1/2,1/3,1/4,1/2,1]\n",
    "V = [Rational(1,5),1,Rational(1,2),Rational(1,3),Rational(1,4),Rational(1,2),1]\n",
    "VA = diag(*V)\n",
    "VA"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}0 & 1 & \\frac{1}{2} & 0 & \\frac{1}{4} & \\frac{1}{2} & 0\\\\\\frac{1}{5} & 0 & \\frac{1}{2} & \\frac{1}{3} & 0 & 0 & 0\\\\\\frac{1}{5} & 0 & 0 & \\frac{1}{3} & \\frac{1}{4} & 0 & 0\\\\\\frac{1}{5} & 0 & 0 & 0 & \\frac{1}{4} & 0 & 0\\\\\\frac{1}{5} & 0 & 0 & \\frac{1}{3} & 0 & \\frac{1}{2} & 1\\\\0 & 0 & 0 & 0 & \\frac{1}{4} & 0 & 0\\\\\\frac{1}{5} & 0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "⎡ 0   1  1/2   0   1/4  1/2  0⎤\n",
       "⎢                             ⎥\n",
       "⎢1/5  0  1/2  1/3   0    0   0⎥\n",
       "⎢                             ⎥\n",
       "⎢1/5  0   0   1/3  1/4   0   0⎥\n",
       "⎢                             ⎥\n",
       "⎢1/5  0   0    0   1/4   0   0⎥\n",
       "⎢                             ⎥\n",
       "⎢1/5  0   0   1/3   0   1/2  1⎥\n",
       "⎢                             ⎥\n",
       "⎢ 0   0   0    0   1/4   0   0⎥\n",
       "⎢                             ⎥\n",
       "⎣1/5  0   0    0    0    0   0⎦"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "trans_mat = Mt*VA\n",
    "trans_mat"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[(0, 1, [Matrix([\n",
      "[ 0],\n",
      "[ 1],\n",
      "[ 0],\n",
      "[ 0],\n",
      "[ 0],\n",
      "[-2],\n",
      "[ 1]])]), (1, 1, [Matrix([\n",
      "[    5],\n",
      "[52/19],\n",
      "[44/19],\n",
      "[33/19],\n",
      "[56/19],\n",
      "[14/19],\n",
      "[    1]])])]\n"
     ]
    }
   ],
   "source": [
    "trans_mat.eigenvects(error_when_incomplete=False)\n",
    "v_all = trans_mat.eigenvects(error_when_incomplete=False)\n",
    "print(v_all)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix([[5], [52/19], [44/19], [33/19], [56/19], [14/19], [1]])\n"
     ]
    }
   ],
   "source": [
    "v = Matrix(trans_mat.eigenvects(error_when_incomplete=False)[1][2])\n",
    "print(v)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.699456533837389\n",
      "0.382860418521518\n",
      "0.323958815672054\n",
      "0.242969111754040\n",
      "0.412311219946250\n",
      "0.103077804986563\n"
     ]
    }
   ],
   "source": [
    "v_n = v.norm().evalf()\n",
    "for i in range(0,6):\n",
    "    print(v[i]/v_n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3.8.5 ('base')",
   "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": "3.8.5"
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  "toc": {
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