05.1-转置和互换.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing(use_latex='mathjax')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 列移动"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  2  3⎤\n",
       "⎢       ⎥\n",
       "⎢4  5  6⎥\n",
       "⎢       ⎥\n",
       "⎣7  8  9⎦"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 0\\\\0 & 0 & 1\\\\1 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  1  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  0  1⎥\n",
       "⎢       ⎥\n",
       "⎣1  0  0⎦"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = Matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 1 & 2\\\\6 & 4 & 5\\\\9 & 7 & 8\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  1  2⎤\n",
       "⎢       ⎥\n",
       "⎢6  4  5⎥\n",
       "⎢       ⎥\n",
       "⎣9  7  8⎦"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Permutation Matrix"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "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": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n = 3\n",
    "eye(n)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Permutation Matrix 是 Indentify  Matrix 行互换的组合, 其可能数量为$n!$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Permutation Matrix 记为P矩阵"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### P矩阵有性质:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (1). $ P^T = P^-1 $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 0\\\\0 & 0 & 1\\\\1 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  1  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  0  1⎥\n",
       "⎢       ⎥\n",
       "⎣1  0  0⎦"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = Matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 1\\\\1 & 0 & 0\\\\0 & 1 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  0  1⎤\n",
       "⎢       ⎥\n",
       "⎢1  0  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  1  0⎦"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.inv()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 1\\\\1 & 0 & 0\\\\0 & 1 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  0  1⎤\n",
       "⎢       ⎥\n",
       "⎢1  0  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  1  0⎦"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### (2). $ P * P^T = I$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 0\\\\0 & 0 & 1\\\\1 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  1  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  0  1⎥\n",
       "⎢       ⎥\n",
       "⎣1  0  0⎦"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = Matrix([[0, 1, 0], [0, 0, 1], [1, 0, 0]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "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": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.T * B"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Symmetric Matrix\n",
    "对称矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 1 & 7\\\\1 & 2 & 9\\\\7 & 9 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  1  7⎤\n",
       "⎢       ⎥\n",
       "⎢1  2  9⎥\n",
       "⎢       ⎥\n",
       "⎣7  9  1⎦"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[3, 1, 7], [1, 2, 9], [7, 9, 1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & 1 & 7\\\\1 & 2 & 9\\\\7 & 9 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  1  7⎤\n",
       "⎢       ⎥\n",
       "⎢1  2  9⎥\n",
       "⎢       ⎥\n",
       "⎣7  9  1⎦"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 矩阵和逆的积为对称矩阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 3\\\\2 & 3\\\\4 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  3⎤\n",
       "⎢    ⎥\n",
       "⎢2  3⎥\n",
       "⎢    ⎥\n",
       "⎣4  1⎦"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, 3], [2, 3], [4, 1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}21 & 13\\\\13 & 19\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡21  13⎤\n",
       "⎢      ⎥\n",
       "⎣13  19⎦"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A.T * A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}10 & 11 & 7\\\\11 & 13 & 11\\\\7 & 11 & 17\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡10  11  7 ⎤\n",
       "⎢          ⎥\n",
       "⎢11  13  11⎥\n",
       "⎢          ⎥\n",
       "⎣7   11  17⎦"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A * A.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 转置分配律:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$(A * B) ^ T = B^T * A^T $"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 3\\\\2 & 3\\\\4 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  3⎤\n",
       "⎢    ⎥\n",
       "⎢2  3⎥\n",
       "⎢    ⎥\n",
       "⎣4  1⎦"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, 3], [2, 3], [4, 1]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 3 & 5\\\\2 & 4 & 6\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  3  5⎤\n",
       "⎢       ⎥\n",
       "⎣2  4  6⎦"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = Matrix([[1, 3, 5], [2, 4, 6]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}7 & 8 & 6\\\\15 & 18 & 16\\\\23 & 28 & 26\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡7   8   6 ⎤\n",
       "⎢          ⎥\n",
       "⎢15  18  16⎥\n",
       "⎢          ⎥\n",
       "⎣23  28  26⎦"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A * B).T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}7 & 8 & 6\\\\15 & 18 & 16\\\\23 & 28 & 26\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡7   8   6 ⎤\n",
       "⎢          ⎥\n",
       "⎢15  18  16⎥\n",
       "⎢          ⎥\n",
       "⎣23  28  26⎦"
      ]
     },
     "execution_count": 48,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.T * A.T"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 逆分配律:\n",
    "$ (A * B) ^ -1 = B ^ -1 * A^ -1 $ <br>\n",
    "注意: A和B都需要是方阵"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 3 & 4\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  3  4⎤\n",
       "⎢       ⎥\n",
       "⎢4  5  6⎥\n",
       "⎢       ⎥\n",
       "⎣7  8  9⎦"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])\n",
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 3 & 4\\\\5 & 6 & 7\\\\8 & 9 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  3  4⎤\n",
       "⎢       ⎥\n",
       "⎢5  6  7⎥\n",
       "⎢       ⎥\n",
       "⎣8  9  0⎦"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B = Matrix([[2, 3, 4], [5, 6, 7], [8, 9, 0]])\n",
    "B"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{23}{5} & - \\frac{173}{15} & \\frac{169}{30}\\\\- \\frac{21}{5} & \\frac{161}{15} & - \\frac{79}{15}\\\\\\frac{3}{5} & - \\frac{28}{15} & \\frac{29}{30}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       -173   169 ⎤\n",
       "⎢23/5   ─────  ─── ⎥\n",
       "⎢         15    30 ⎥\n",
       "⎢                  ⎥\n",
       "⎢        161   -79 ⎥\n",
       "⎢-21/5   ───   ────⎥\n",
       "⎢         15    15 ⎥\n",
       "⎢                  ⎥\n",
       "⎢       -28     29 ⎥\n",
       "⎢ 3/5   ────    ── ⎥\n",
       "⎣        15     30 ⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(A * B).inv()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{23}{5} & - \\frac{173}{15} & \\frac{169}{30}\\\\- \\frac{21}{5} & \\frac{161}{15} & - \\frac{79}{15}\\\\\\frac{3}{5} & - \\frac{28}{15} & \\frac{29}{30}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡       -173   169 ⎤\n",
       "⎢23/5   ─────  ─── ⎥\n",
       "⎢         15    30 ⎥\n",
       "⎢                  ⎥\n",
       "⎢        161   -79 ⎥\n",
       "⎢-21/5   ───   ────⎥\n",
       "⎢         15    15 ⎥\n",
       "⎢                  ⎥\n",
       "⎢       -28     29 ⎥\n",
       "⎢ 3/5   ────    ── ⎥\n",
       "⎣        15     30 ⎦"
      ]
     },
     "execution_count": 57,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "B.inv() * A.inv()"
   ]
  }
 ],
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