SymPy_Solvers_Linear_Algebra.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()\n",
    "x,y,z=symbols('x y z')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving equations\n",
    "__NOTE__: Symbolic equations in SymPy are not represented by __`=`__ or __`==`__, but by __`Eq`__."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAANBAMAAAD2/0vEAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHarIkSJZt3NuzJUme9mZ+xlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArElEQVQYGWNgEFJ2VVNgQAGMugoMfgIMjAnsTRwTgDJsM8HgAojJ+oBBH0gJsH1kdUDRxODDO4FhFQMDIwPXAlQJBgYBPgWGTyBBfgV0KYZ4Ae7fIMH9G8BSULsMQJybDEwPGBh4BPQZmATAkghiMQMXUEn8gfUM7ghBCOsyQ3wBA4OwktB1IIUKhJXuO6CKIPFuIrGRmcwTGL8i85HYHAVcCkhcZCar0m0gFwByxSGozdHRGQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$x = y$$"
      ],
      "text/plain": [
       "x = y"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Eq(x, y)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In SymPy, any expression not in an `Eq` is automatically assumed to equal 0 by the solving functions. Since `a=b` if and only if `a−b=0`, this means that instead of using `x == y`, you can just use `x - y`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEAAAAAVBAMAAAAEBPWpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQzYndq1TvMrsiRHaZuvfyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVQoFWNgYGBUZMABFj0GSTC74pAGCt9cACQYP+BWwOQAlGNNABJw0GkAZXJMBjLAcsgKmEvroQrMwz9jU8DAIA8zgZFKCiYA7WF3ABJwgGoF2IedC+CyQAaqAgYnBobVOUBhZhcwACpFU8CeBw0tuCFoCtyBEuwb4LJABqoCsBuQAwpdASvIF6gK3hswMH8BGcmHPaDU5zsJMKQBQ5Op8gswFWDEBUgrAwMHUAEEQBSA7EEBrHAeWAFmerCDK2ByADExUpQASBQMbl4AURx40yQA7h8x6DqDyqQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left\\{-1, 1\\right\\}$$"
      ],
      "text/plain": [
       "{-1, 1}"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(Eq(x**2, 1), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{-1, 1\\right\\}$$"
      ],
      "text/plain": [
       "{-1, 1}"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(Eq(x**2 - 1, 0), x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAEAAAAAVBAMAAAAEBPWpAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQzYndq1TvMrsiRHaZuvfyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA1ElEQVQoFWNgYGBUZMABFj0GSTC74pAGCt9cACQYP+BWwOQAlGNNABJw0GkAZXJMBjLAcsgKmEvroQrMwz9jU8DAIA8zgZFKCiYA7WF3ABJwgGoF2IedC+CyQAaqAgYnBobVOUBhZhcwACpFU8CeBw0tuCFoCtyBEuwb4LJABqoCsBuQAwpdASvIF6gK3hswMH8BGcmHPaDU5zsJMKQBQ5Op8gswFWDEBUgrAwMHUAEEQBSA7EEBrHAeWAFmerCDK2ByADExUpQASBQMbl4AURx40yQA7h8x6DqDyqQAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left\\{-1, 1\\right\\}$$"
      ],
      "text/plain": [
       "{-1, 1}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(x**2 - 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving Equations Algebraically\n",
    "The main function for solving algebraic equations is __`solveset`__. The syntax for solveset is __`solveset(equation, variable=None, domain=S.Complexes)`__ where equations may be in the form of `Eq` instances or expressions that are assumed to be equal to zero."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{0, 1\\right\\}$$"
      ],
      "text/plain": [
       "{0, 1}"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(x**2 - x, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAA8AAAAOBAMAAAACpFvcAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZt3vu80QqzIiiZlEVHY0fSkZAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAXElEQVQIHWMQMvEPNRFgAIKAdIYWkQQgw2EWAwtnAYTBPSEAxnCAMNh5oFKXtiaARCYrqV4A0kBdp1lAAkAG9wkHCIPBTQ7KaOduALKAVnBdqGBgAFoaweASfwEAz5AZQH/RFQAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\mathbb{R}$$"
      ],
      "text/plain": [
       "ℝ"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(x - x, x, domain=S.Reals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{2 n \\pi + \\frac{\\pi}{2}\\; |\\; n \\in \\mathbb{Z}\\right\\}$$"
      ],
      "text/plain": [
       "⎧        π        ⎫\n",
       "⎨2⋅n⋅π + ─ | n ∊ ℤ⎬\n",
       "⎩        2        ⎭"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(sin(x) - 1, x, domain=S.Reals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{-1, 1\\right\\}$$"
      ],
      "text/plain": [
       "{-1, 1}"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(x**4 - 1, x, domain=S.Reals)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAHEAAAAVBAMAAACDNRyvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQzYndq1TvMrsiRHaZuvfyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABR0lEQVQ4Eb1UvUrDUBg9Ma0J2kp8glpwLh0Et7aDLk6ZKm4FwcnBSXArOLkJboLQ3cXX6WQfQbCz+P16ExOLdfAO95x7vnOSm+TLBRB1sebY6vU5ER+vmSN7+5Ez0dsvk8mLGhn3mTYnKsicPvgiMFeSTBnjmGkxeXj6br7ATChDJYnIkwVWzujqn5KDnt5O8Ix5MlJF5hW7zefqE+wwv5upIvPPydZUvgQUG3vA8yVFDo54nNDHrb4hq8X4kGsbXi+A+UwUnWqSXk3tooKNK5K9NcSxIrnNZhqC8pzFTqjbrfqB4TQXKijvtpTcoS0NR+wITPzAayuLl8QJa7pv42bZxeY51QOzIJJd4KJPT0dY7Vt13boZgamUUlKGdt+9rb4gr2EqNb0iycr/GS+8HpgpA69oV3w/EyIvIzCTMsP2k5D0b+fQJ9iESF/AIGy8AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left\\{-1, 1, - i, i\\right\\}$$"
      ],
      "text/plain": [
       "{-1, 1, -ⅈ, ⅈ}"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solveset(x**4 - 1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In the `solveset` module, the linear system of equations is solved using `linsolve`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Linear system form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{\\left ( - y - 1, \\quad y, \\quad 2\\right )\\right\\}$$"
      ],
      "text/plain": [
       "{(-y - 1, y, 2)}"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve([x + y + z - 1, x + y + 2*z - 3 ], (x, y, z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Augmented matrix form"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAALMAAAAVBAMAAAD/f8IzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQzYndq1TvMrsiRHaZuvfyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACL0lEQVQ4Ea1VPWsUURQ9k9mZXXczy/wAIZMoBCx0CjFIIBnENCIkBCLYhAVBLARtTLugVcBiq4CBkOnTxCpFUkwVtLMSrLLkByQpNIUQk3vv25l5s/MGFtxT3HfOfWfOfL43AKxpjBvN+yFH2ktcJ32u48BnDvG2uVoXXOe5/C/smcMY9YRj7nJxOlyfc6nEptxf5XQ6MQfnCnbE8gUXibYCphWwP3RHi74DfAHWOSaPdhPWlZgaLXoL6CZ4zDF59HFlqkyMGL0cohuj9qMQfZpH26d9fB+6ynK0ycUZ70I4AY0vWdQjKl+ZKTheD+epGIzlaJOLzN4f+uQiIlN85GZM5Q0zhW8TEQ5TMRjL0SYXmd33QKNDpBYAe2+JYIeWzlNBHN7q4x+1Hop8xrMoR6cumc7LDNHmBeuNPvArJkLRGU7C5t9MKFKOhsEFtAPyS3SNLh/1fSraA6GF1O5RS4ch2uACPmIygdehQ+VZy5LRXiMO0OJT6jBEG1xorqCdqNcoX4hEr2pBszjxNcn0PIT9u9gT12JUaB6vrW4AEyvUzJfMT80yd/Yg0STR2d0nPl6Hhaa43FeF3vL19SV9HDE182g3IZlBtq1MDUijGK02t0/DLtKPuKeie8SsgIpCPbKvUq6Pji5ooYmL730Y+fak9uujzOD6rX4mNLKgcVob4rINTrWpqhtf4mPyX4F3dq8Qkgo/JTIql1XoKdFKaPTUQmlMEx/fD+w2pcm/8Qa0F2zX0R1YWgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left\\{\\left ( - y - 1, \\quad y, \\quad 2\\right )\\right\\}$$"
      ],
      "text/plain": [
       "{(-y - 1, y, 2)}"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "linsolve(Matrix(([1, 1, 1, 1], [1, 1, 2, 3])), (x, y, z))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### In the `solveset` module, a non linear system of equations is solved using `nonlinsolve`. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{\\left ( - i, \\quad - \\sqrt{- z}, \\quad z\\right ), \\left ( - i, \\quad \\sqrt{- z}, \\quad z\\right ), \\left ( i, \\quad - \\sqrt{- z}, \\quad z\\right ), \\left ( i, \\quad \\sqrt{- z}, \\quad z\\right ), \\left ( x, \\quad - \\sqrt{- z}, \\quad z\\right ), \\left ( x, \\quad \\sqrt{- z}, \\quad z\\right )\\right\\}$$"
      ],
      "text/plain": [
       "⎧⎛       ____   ⎞  ⎛      ____   ⎞  ⎛      ____   ⎞  ⎛     ____   ⎞  ⎛      __\n",
       "⎨⎝-ⅈ, -╲╱ -z , z⎠, ⎝-ⅈ, ╲╱ -z , z⎠, ⎝ⅈ, -╲╱ -z , z⎠, ⎝ⅈ, ╲╱ -z , z⎠, ⎝x, -╲╱ -\n",
       "⎩                                                                             \n",
       "\n",
       "__   ⎞  ⎛     ____   ⎞⎫\n",
       "z , z⎠, ⎝x, ╲╱ -z , z⎠⎬\n",
       "                      ⎭"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nonlinsolve([x**2 + 1, y**2 + z], [x, y, z])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left\\{\\left ( \\frac{1}{4} - \\frac{\\sqrt{103} i}{4}, \\quad \\frac{3}{8} + \\frac{\\sqrt{103} i}{8}\\right ), \\left ( \\frac{1}{4} + \\frac{\\sqrt{103} i}{4}, \\quad \\frac{3}{8} - \\frac{\\sqrt{103} i}{8}\\right )\\right\\}$$"
      ],
      "text/plain": [
       "⎧⎛1   √103⋅ⅈ  3   √103⋅ⅈ⎞  ⎛1   √103⋅ⅈ  3   √103⋅ⅈ⎞⎫\n",
       "⎨⎜─ - ──────, ─ + ──────⎟, ⎜─ + ──────, ─ - ──────⎟⎬\n",
       "⎩⎝4     4     8     8   ⎠  ⎝4     4     8     8   ⎠⎭"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "nonlinsolve([x**2+y+6,2*y+x-1],[x,y])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Currently `nonlinsolve` is not properly capable of solving the system of equations having trigonometric functions. Use `solve` instead"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ \\left ( - \\frac{\\pi}{4}, \\quad \\frac{3 \\pi}{4}\\right ), \\quad \\left ( \\frac{3 \\pi}{4}, \\quad - \\frac{\\pi}{4}\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛-π   3⋅π⎞  ⎛3⋅π  -π ⎞⎤\n",
       "⎢⎜───, ───⎟, ⎜───, ───⎟⎥\n",
       "⎣⎝ 4    4 ⎠  ⎝ 4    4 ⎠⎦"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve([sin(x + y)-1, cos(x - y)+1], [x, y])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ - i \\log{\\left (1 + \\sqrt{2} \\right )}, \\quad i \\log{\\left (1 + \\sqrt{2} \\right )}, \\quad \\pi - i \\log{\\left (1 + \\sqrt{2} \\right )}, \\quad \\pi + i \\log{\\left (1 + \\sqrt{2} \\right )}\\right ]$$"
      ],
      "text/plain": [
       "[-ⅈ⋅log(1 + √2), ⅈ⋅log(1 + √2), π - ⅈ⋅log(1 + √2), π + ⅈ⋅log(1 + √2)]"
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "solve(sin(x)**2+1, x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Solving Differential Equations\n",
    "First, create an undefined function by passing __`cls=Function`__ to the symbols function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [],
   "source": [
    "f, g = symbols('f g', cls=Function)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`f` and `g` are now undefined functions. We can call `f(x)`, and it will represent an unknown function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "f(x)"
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Derivatives of `f(x)` are unevaluated."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\frac{d}{d x} f{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "d       \n",
       "──(f(x))\n",
       "dx      "
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f(x).diff(x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To represent the differential equation $f^{′′}(x)−2f^′(x)+f(x)=sin(x)$, we would thus use"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$f{\\left (x \\right )} - 2 \\frac{d}{d x} f{\\left (x \\right )} + \\frac{d^{2}}{d x^{2}} f{\\left (x \\right )} = \\sin{\\left (x \\right )}$$"
      ],
      "text/plain": [
       "                      2               \n",
       "         d           d                \n",
       "f(x) - 2⋅──(f(x)) + ───(f(x)) = sin(x)\n",
       "         dx           2               \n",
       "                    dx                "
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diffeq = Eq(f(x).diff(x, x) - 2*f(x).diff(x) + f(x), sin(x))\n",
    "diffeq"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To solve the ODE, pass it and the function to solve for to __`dsolve`__. Remember to solve for `f(x` not for `x`"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$f{\\left (x \\right )} = \\left(C_{1} + C_{2} x\\right) e^{x} + \\frac{\\cos{\\left (x \\right )}}{2}$$"
      ],
      "text/plain": [
       "                    x   cos(x)\n",
       "f(x) = (C₁ + C₂⋅x)⋅ℯ  + ──────\n",
       "                          2   "
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dsolve(diffeq, f(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`dsolve` returns an instance of `Eq`. This is because in general, solutions to differential equations cannot be solved explicitly for the function. The arbitrary constants in the solutions from dsolve are symbols of the form `C1`, `C2`, `C3`, and so on."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & -1\\\\3 & 4\\\\0 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  -1⎤\n",
       "⎢     ⎥\n",
       "⎢3  4 ⎥\n",
       "⎢     ⎥\n",
       "⎣0  2 ⎦"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix([[1, -1], [3, 4], [0, 2]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1\\\\2\\\\3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1⎤\n",
       "⎢ ⎥\n",
       "⎢2⎥\n",
       "⎢ ⎥\n",
       "⎣3⎦"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Matrix([1, 2, 3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAyBAMAAACqpzYoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUIhAyzWaZq3a7RN1A9Q5DAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAx0lEQVQoFWOQ//+JAQT0/39kEHZxBbNDXAwZxMAsEJEIZlcEZBrA2BA9EPGi1Q5wNQuALJgaZPaK7mlw8aMM/hdgahgYeB4g2JxfoGyuAgamv1A2YwED51com5OBga8AymaZwNDoAGUzeGpNhZsPZAw420npMMwNQHfaN0Ddyf2Zgf8BlM0ry8AP8wtQLVwNAwPrI5heBhY9ZH9tRAo3RlFYODgw8HyGsvm/ItiMBxj4YOHG0YBk/s2ZZ+HmAxkUs5HTDFJaAgCuIUOfsBnT3QAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}5\\\\3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡5⎤\n",
       "⎢ ⎥\n",
       "⎣3⎦"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[1, 2, 3], [3, 2, 1]])\n",
    "N = Matrix([0, 1, 1])\n",
    "M*N"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Shape, indexing, deleting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2 & 3\\\\-2 & 0 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   2  3⎤\n",
       "⎢        ⎥\n",
       "⎣-2  0  4⎦"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[1, 2, 3], [-2, 0, 4]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 2, \\quad 3\\right )$$"
      ],
      "text/plain": [
       "(2, 3)"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.shape"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[1  2  3]"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.row(0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABYAAAAyBAMAAACqpzYoAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUIhAyu5lmdt1EzataiJGiAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVQoFWOQ//+JAQT0/39kEHZxBbNDXAwZxMAsEJEIZjspFcPYLA0M9hOg4lyfGfg3QNk8Mgz8F2DqGRBqGBhYN8H0MrBot8LZDAwLDyDUM4pC2SwODOyfoWz+rwg2YwED91eoOOcEJPNPdNQgmQlkQt0MYg0MOwPJ3n0INpccgu15D8FWQLA5HRDsKAYEWwHBZk5AsIMYEGy7jo6PXQj/yiHMZxBHsNX/v0SoAQpTGFbIaQYpLQEAudM/WBAuQk4AAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3\\\\4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3⎤\n",
       "⎢ ⎥\n",
       "⎣4⎦"
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.col(-1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADUAAAAyBAMAAAAOzY77AAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMA74lUIhAymburRN3NdmbDbtPBAAAACXBIWXMAAA7EAAAOxAGVKw4bAAABaElEQVQ4Ee3Uu07DMBQG4D9Jk7ZpKEFIgFiIOiA2MjEjKGIksDB0APEE2VomOrE2E6zhCahgQEwg8QJFQmLiMjAiBqBIiItxRewcQ9SZAU8+/mLL59gxJtgjMprOmIvh6kIGwazOuxjJkt6Y9W2rjW36Ra4y5QszI0zHBBdhfArLB9BcYjc+3oRpbRSfie3FeBfmdFF8IQayJh8udRWreWIeH97nQdrmjgJixyn0egPjqeVD1TAUiNxxqVAVaPnCjBA8lo0FxDaACynAGHAi1swdVDbbxB5Ifg5jjJpdb8TpPskk2U3OSMa082/QrydpRXgt+VkndTlDbVbBUiSs7EI/VexcWiGCodzB3L20wQj2E52nO9LWQtgf1O5SW/JgUTM7xEJY9MfWkdrPNWeI8b0YZC+mR6zQRJnkYOzWt0Y7SV1+567JHHCI5XWaA7SmqBlWrnYU0luvt6LWCojgj93Pfm9kn7f1CwvjWhKmF20sAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 3\\\\0 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  3⎤\n",
       "⎢    ⎥\n",
       "⎣0  4⎦"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.col_del(0)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAADMAAAAZBAMAAABuuAtJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAq90iEJm7MolE7812VGYvQfzlAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAyklEQVQoFWMQUjJgwAImKTUwKGARBwkxCxCQYglNK0DWa9GRC9PlyMDxE0mKOYDh/gOogSEMDKuQpHj+MvArQKUWMTDIIzmTcyUD/wao1H4HBnkUyxAGAg3Td0AykYFFA+YMBgbOH8gyzE1A22H+YhJAlmJgODUBLhWKKsPAtQQmxROALMVswMD2FyZ1lQHIhQP+Lwgp9gYGHiQprgQGpi9QXUYdrUJwPQwMfA8Q/tr///8nJCmGqaGZCMcjS0DZMH8NaiklXOlQAAA95imi07ZmXQAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "[2  3]"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.row_del(1)\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}2 & 3\\\\0 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡2  3⎤\n",
       "⎢    ⎥\n",
       "⎣0  4⎦"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = M.row_insert(1, Matrix([[0, 4]]))\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2 & 3\\\\-2 & 0 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   2  3⎤\n",
       "⎢        ⎥\n",
       "⎣-2  0  4⎦"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = M.col_insert(0, Matrix([1, -2]))\n",
    "M"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Basic operations, special matrices"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [],
   "source": [
    "M = Matrix([[1, 3], [-2, 3]])\n",
    "N = Matrix([[0, 3], [0, 7]])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 6\\\\-2 & 10\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   6 ⎤\n",
       "⎢      ⎥\n",
       "⎣-2  10⎦"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M+N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0\\\\-2 & -4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   0 ⎤\n",
       "⎢      ⎥\n",
       "⎣-2  -4⎦"
      ]
     },
     "execution_count": 63,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M-N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 64,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 24\\\\0 & 15\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  24⎤\n",
       "⎢     ⎥\n",
       "⎣0  15⎦"
      ]
     },
     "execution_count": 64,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M*N"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-5 & 12\\\\-8 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-5  12⎤\n",
       "⎢      ⎥\n",
       "⎣-8  3 ⎦"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 67,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}\\frac{1}{3} & - \\frac{1}{3}\\\\\\frac{2}{9} & \\frac{1}{9}\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1/3  -1/3⎤\n",
       "⎢         ⎥\n",
       "⎣2/9  1/9 ⎦"
      ]
     },
     "execution_count": 67,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M**-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Matrix det == 0; not invertible.\n"
     ]
    }
   ],
   "source": [
    "try: N**-1\n",
    "except ValueError as e: print(e)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  2  3⎤\n",
       "⎢       ⎥\n",
       "⎣4  5  6⎦"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[1, 2, 3], [4, 5, 6]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 4\\\\2 & 5\\\\3 & 6\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  4⎤\n",
       "⎢    ⎥\n",
       "⎢2  5⎥\n",
       "⎢    ⎥\n",
       "⎣3  6⎦"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.T"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\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": 70,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "eye(3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 1 & 1 & 1\\\\1 & 1 & 1 & 1\\\\1 & 1 & 1 & 1\\\\1 & 1 & 1 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  1  1  1⎤\n",
       "⎢          ⎥\n",
       "⎢1  1  1  1⎥\n",
       "⎢          ⎥\n",
       "⎢1  1  1  1⎥\n",
       "⎢          ⎥\n",
       "⎣1  1  1  1⎦"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ones(4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  0  0  0  0⎤\n",
       "⎢             ⎥\n",
       "⎢0  0  0  0  0⎥\n",
       "⎢             ⎥\n",
       "⎢0  0  0  0  0⎥\n",
       "⎢             ⎥\n",
       "⎢0  0  0  0  0⎥\n",
       "⎢             ⎥\n",
       "⎣0  0  0  0  0⎦"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zeros(5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 2 & 0\\\\0 & 0 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0  0⎤\n",
       "⎢       ⎥\n",
       "⎢0  2  0⎥\n",
       "⎢       ⎥\n",
       "⎣0  0  4⎦"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diag(1,2,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-1 & 0 & 0 & 0\\\\0 & 1 & 1 & 0\\\\0 & 1 & 1 & 0\\\\0 & 0 & 0 & 5\\\\0 & 0 & 0 & 7\\\\0 & 0 & 0 & 5\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-1  0  0  0⎤\n",
       "⎢           ⎥\n",
       "⎢0   1  1  0⎥\n",
       "⎢           ⎥\n",
       "⎢0   1  1  0⎥\n",
       "⎢           ⎥\n",
       "⎢0   0  0  5⎥\n",
       "⎢           ⎥\n",
       "⎢0   0  0  7⎥\n",
       "⎢           ⎥\n",
       "⎣0   0  0  5⎦"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diag(-1, ones(2, 2), Matrix([5, 7, 5]))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Determinant"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAGkAAABLCAMAAABEOFk1AAAAPFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMAMquZdlQiEEDpMETviWbN3btsl8C/KgAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAuVJREFUWAntmemWnCAQhVFQMhFcwvu/a1hc6go2Nd3Tc04m+qMbWeqTK8JFRePi0Yp3HXMCCNE4qfzRvQskxhC+dYHUvw1yBO6eI1k12uaIwkmdSZ3RjGaTV1rPDCVIOCDp1raOQRpNuBo7Va4JwwHJtxw5pCUO045RlYR7iuQiaXD10foiSTsZdBucqsgHEj3Tp97ZgGi+gxT71CTgw379M+qJNCK6948IsSxBMqLMpYKkzjMjQjCfXLyaM0kxHkch5jAbGcbER8IhSbbGTXK8FGMr0NavAnUQhEPSFukd/zfpFVVv9W71UIGfPyK0lMtSn89Qluuz3lq7bLYG1NPB81g3XLc9lxDneC4Sog8moHPrhA0kGV2lib4xb5jloHPMioWKi7KbUwmQTMxsP7EhIGtqThpMUMetVhdIU1wGW8e/Uw9JkV1WL13WzFp2U90qSc/RG/ra0KfYemDYuITxvxVSb+d074uk+TO70ArJA1S8WSWSjP7Ul+SHnsiR7mWdJMy6Iwvq/fr4vYdV16C9Dkk8JPVxK7esBvTPB+xzuwBKNUi86+RDkotDayPhiGjiQFGMbeUKf0gy0epuQxlI/ewnKtmuD/V1T44S4hyPzC01hnmo/DxN6Q0FmwTOcYtP/gfpZ+xtFoU+kUpfn7xJr2h6q3erhwr8HyOiYaxQYBlRJHoGRjVXj7G6o2WksSGNRjUjKVP3EWgZITo9QaN6JvXjVCehZaTBIY1G9UyygkGK8fYlDqLTEzSqJ9LYc0mHZaTBC+ni6i60f9/J6hO1jIXoJGs3qtinYFhYJF9vt4wkbCG5D2UgdcEuckm7ZSyEP7IOo0pJOnowBgkt4xE2TxGjSkld672MdEbW3k6jZczj7znUqFJSqsB4ctEy7nGzBBjVAik6z6wVzUDLSEsgjUb1TLKLc231pSVYRohOT9Conkm05temb9Irev5c9b7ry7EOH3aV2rZUr9yMctv45Vgp8RfX+ib5WYdU3gAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 1\\\\2 & -1 & 3\\\\4 & 3 & 2\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  0   1⎤\n",
       "⎢        ⎥\n",
       "⎢2  -1  3⎥\n",
       "⎢        ⎥\n",
       "⎣4  3   2⎦"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[1, 0, 1], [2, -1, 3], [4, 3, 2]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 80,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABgAAAAPBAMAAAAMihLoAAAAJFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAADHJj5lAAAAC3RSTlMAEM0yVO+riWZ2mSMU5s8AAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAsSURBVAgdY2DAClgCEcJiZRsRHAYOMjlCSiCgzECWAZxIlnKlblqB5B4GBgDVtwt2YFScIgAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$-1$$"
      ],
      "text/plain": [
       "-1"
      ]
     },
     "execution_count": 80,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.det()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Reduced row echelon form\n",
    "To put a matrix into reduced row echelon form, use `rref`. `rref` returns a tuple of two elements. The first is the reduced row echelon form, and the second is a tuple of indices of the pivot columns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 81,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 0 & 1 & 3\\\\2 & 3 & 4 & 7\\\\-1 & -3 & -3 & -4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1   0   1   3 ⎤\n",
       "⎢              ⎥\n",
       "⎢2   3   4   7 ⎥\n",
       "⎢              ⎥\n",
       "⎣-1  -3  -3  -4⎦"
      ]
     },
     "execution_count": 81,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[1, 0, 1, 3], [2, 3, 4, 7], [-1, -3, -3, -4]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left[\\begin{matrix}1 & 0 & 1 & 3\\\\0 & 1 & \\frac{2}{3} & \\frac{1}{3}\\\\0 & 0 & 0 & 0\\end{matrix}\\right], \\quad \\left ( 0, \\quad 1\\right )\\right )$$"
      ],
      "text/plain": [
       "⎛⎡1  0   1    3 ⎤        ⎞\n",
       "⎜⎢              ⎥        ⎟\n",
       "⎜⎢0  1  2/3  1/3⎥, (0, 1)⎟\n",
       "⎜⎢              ⎥        ⎟\n",
       "⎝⎣0  0   0    0 ⎦        ⎠"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.rref()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Nullspace and columnspace"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 83,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 2 & 3 & 0 & 0\\\\4 & 10 & 0 & 0 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  2   3  0  0⎤\n",
       "⎢              ⎥\n",
       "⎣4  10  0  0  1⎦"
      ]
     },
     "execution_count": 83,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[1, 2, 3, 0, 0], [4, 10, 0, 0, 1]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 84,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}-15\\\\6\\\\1\\\\0\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\0\\\\0\\\\1\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\- \\frac{1}{2}\\\\0\\\\0\\\\1\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡-15⎤  ⎡0⎤  ⎡ 1  ⎤⎤\n",
       "⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥\n",
       "⎢⎢ 6 ⎥  ⎢0⎥  ⎢-1/2⎥⎥\n",
       "⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥\n",
       "⎢⎢ 1 ⎥, ⎢0⎥, ⎢ 0  ⎥⎥\n",
       "⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥\n",
       "⎢⎢ 0 ⎥  ⎢1⎥  ⎢ 0  ⎥⎥\n",
       "⎢⎢   ⎥  ⎢ ⎥  ⎢    ⎥⎥\n",
       "⎣⎣ 0 ⎦  ⎣0⎦  ⎣ 1  ⎦⎦"
      ]
     },
     "execution_count": 84,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.nullspace()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}1 & 1 & 2\\\\2 & 1 & 3\\\\3 & 1 & 4\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡1  1  2⎤\n",
       "⎢       ⎥\n",
       "⎢2  1  3⎥\n",
       "⎢       ⎥\n",
       "⎣3  1  4⎦"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[1, 1, 2], [2 ,1 , 3], [3 , 1, 4]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ \\left[\\begin{matrix}1\\\\2\\\\3\\end{matrix}\\right], \\quad \\left[\\begin{matrix}1\\\\1\\\\1\\end{matrix}\\right]\\right ]$$"
      ],
      "text/plain": [
       "⎡⎡1⎤  ⎡1⎤⎤\n",
       "⎢⎢ ⎥  ⎢ ⎥⎥\n",
       "⎢⎢2⎥, ⎢1⎥⎥\n",
       "⎢⎢ ⎥  ⎢ ⎥⎥\n",
       "⎣⎣3⎦  ⎣1⎦⎦"
      ]
     },
     "execution_count": 86,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.columnspace()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Eigenvalues, Eigenvectors, Diagonalization, and Characteristic polynomial"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & -2 & 4 & -2\\\\5 & 3 & -3 & -2\\\\5 & -2 & 2 & -2\\\\5 & -2 & -3 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  -2  4   -2⎤\n",
       "⎢             ⎥\n",
       "⎢5  3   -3  -2⎥\n",
       "⎢             ⎥\n",
       "⎢5  -2  2   -2⎥\n",
       "⎢             ⎥\n",
       "⎣5  -2  -3  3 ⎦"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M = Matrix([[3, -2,  4, -2], [5,  3, -3, -2], [5, -2,  2, -2], [5, -2, -3,  3]])\n",
    "M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 88,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAANAAAAAVBAMAAADFr31AAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQzYndq1TvMrsiRHaZuvfyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAACP0lEQVRIDb1WMWgUQRR9u5u7vTWXu7Q2cjkLQZvDJqUb8bDMEThTZkGCjZizEWyMaGVlQBFSiEvAyiZgI0rgEJJK8LBLlRArwUI4lSCG9c/szM4c9/e2ih9u9v3/3/y3M/Pn7gA4czhl2/oqBLz2KctQ+f2YBueHJXT50nvL4+Bu9x0XtmILra2Hmes1P8SAGwIoRTQo83pYGWgHqDwzWKFqA09nTZRh4FXyyxDmUTpRGraQ20KwkbHml4cZ1qB2jHqkHXAMtB/0MwLOA5uMUBDB/2tYzrjQzEvUw4kM9EwaeAGs99Nds1c0NYT/x/AYIUraW8cxRoQWW1iPx4WoTFmu4pZU48rAW5W5fMbdcxctBrDWQkmchx/KsB4OOoS859LlhKo3aNsnMt7i88AwMEOtITv7SSwmZpa+aepyQsA3UUYZz5iOdJ6e5R4NV4E3t+npXZMWU8c3yNXGlwmspucZrtVQaIpq/h11a3VpHGWIAFOm2sf0seEwDDoKcXW01RoCXaePvy2QskoDnzRmherDIqEghGtdi0egd5NnZLc3XgPfJwoF26ovUxazIpeOJcxqnOmgRvdogwK2UHW1eyGimOq6Gr3ZlZB8Y85seo/yGfQldbZPE1LGXnfpntKwhaaSJIlEVdl77v2fcyjfFL6xL80d6eQzPi7LC5DWWEyS34yQqWfQYwNzUDFDLkbuX04JEe5MyKWpYoYUGvk9Gi/qHY7HRiPFDLihmNIWQ645uRmdKGZgfyDIlf/yn+EfoeCE3q0lTOgAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left \\{ -2 : 1, \\quad 3 : 1, \\quad 5 : 2\\right \\}$$"
      ],
      "text/plain": [
       "{-2: 1, 3: 1, 5: 2}"
      ]
     },
     "execution_count": 88,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ \\left ( -2, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}0\\\\1\\\\1\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 3, \\quad 1, \\quad \\left [ \\left[\\begin{matrix}1\\\\1\\\\1\\\\1\\end{matrix}\\right]\\right ]\\right ), \\quad \\left ( 5, \\quad 2, \\quad \\left [ \\left[\\begin{matrix}1\\\\1\\\\1\\\\0\\end{matrix}\\right], \\quad \\left[\\begin{matrix}0\\\\-1\\\\0\\\\1\\end{matrix}\\right]\\right ]\\right )\\right ]$$"
      ],
      "text/plain": [
       "⎡⎛       ⎡⎡0⎤⎤⎞  ⎛      ⎡⎡1⎤⎤⎞  ⎛      ⎡⎡1⎤  ⎡0 ⎤⎤⎞⎤\n",
       "⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥\n",
       "⎢⎜       ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥  ⎢-1⎥⎥⎟⎥\n",
       "⎢⎜-2, 1, ⎢⎢ ⎥⎥⎟, ⎜3, 1, ⎢⎢ ⎥⎥⎟, ⎜5, 2, ⎢⎢ ⎥, ⎢  ⎥⎥⎟⎥\n",
       "⎢⎜       ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥⎥⎟  ⎜      ⎢⎢1⎥  ⎢0 ⎥⎥⎟⎥\n",
       "⎢⎜       ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥⎥⎟  ⎜      ⎢⎢ ⎥  ⎢  ⎥⎥⎟⎥\n",
       "⎣⎝       ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣1⎦⎦⎠  ⎝      ⎣⎣0⎦  ⎣1 ⎦⎦⎠⎦"
      ]
     },
     "execution_count": 89,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.eigenvects()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "P, D = M.diagonalize()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0 & 1 & 1 & 0\\\\1 & 1 & 1 & -1\\\\1 & 1 & 1 & 0\\\\1 & 1 & 0 & 1\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0  1  1  0 ⎤\n",
       "⎢           ⎥\n",
       "⎢1  1  1  -1⎥\n",
       "⎢           ⎥\n",
       "⎢1  1  1  0 ⎥\n",
       "⎢           ⎥\n",
       "⎣1  1  0  1 ⎦"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAIgAAABkCAMAAABq4oiZAAAAPFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMAMquZdlQiEEAw7UTNu4nv3WZsVDW/PQAAAAlwSFlzAAAOxAAADsQBlSsOGwAABFNJREFUaAXtW217qyAMpWp1d61v4///1wmKJaeJp27ttvs8+oXEhOQYGXJgdScfr8L90jXM+Z07+bKarvMv4XBjyF74AKT+LQy3vOdNIE1Ztu3p5o1SV43dk8ybQJowbDp/wfxJv05vsxnMcu4ybwIpm5Cy71NiaMdo6K5wO6n7zJtA+iEELawh1Ma/s7OPcFP6W7vPvAnkGnMU3hgGPgK5eOPPbZ95E8j8dIPxyI0vg8PFV7cqZNJOMwdy8V0WPhPr2XAygOw0cyCDNePWc0VOBtCdZgGkuWbXMjDKWP+sEKu4s/Zrv1nA3gII+Aa1MnE4N4/G8/ZgfdRMgJwDjtqYsto2QB2Nsez2mbeBnOI4rQwg+2asADq/oPcmkHrouq4s4rSWx0jyEKb43phlnNtl3gRynRcKJpCmmz7fJg63y7wJJD35T7QHEKzyUZGjIlgB1I8xclQEK4C6MUYIc1qi1NM3sTWWzsGFRBFmHQihRglHWKyc/biodw2JIs0qEFgq3GVYblRxbeatjzOJAmYVCKFGCdilD2TUW0yPRAGzCoRQowQktvarIVHArAHBBbZIDEozGKzHkSho1oAQapRBqbthJurZvSSSKGjWgUQOYTGnlGluqzhQ5L2ofYdgLeGwakqW7Fbf67sBJAqaQ0Xe/L8s8CQS5pScZ77TPodgfWhbV4QaJSA+UisTCIkCZm2MOJhrUmJs+8j0rF0LFgWSqEAYNVoQjWFyt+cRFkXyLx0IoUapNJdy2nU0t/oOgpXq9LVWfzVfi/WtXgcQLN9REa0iTaV/ttD3pXo9bRv/B+c1L62BDP7XB6vgPhJ6rm27XcfGnUqbgIneekUk98lTC5m49WFT0lpbOyd7q0BgqSCSZwpzu3aFtVc8RYHeKhDgPlluITI361hjDgK9VSDAfUT2TGFu20CgtwYEF9hZ7lykbsU4VqW1MY29NSDIffLsmUzd2mnCrq3DWuytA3mIYBEGtUBujc0C7K0BwaplVcjFx9xK47AWewcgb+9fI1iEhxVxv8I8SIfeH+/KRw+4T16GXCZufQRSPnjApb0anGvy5LkMU1JuCnIcaG6w9nGgtwqEUaOUUlKkdDe1dUAy9sY5HB5w6UAeJFjErS67wtw+mQ7hxPmXDiQ91A+2BxAs9lGRoyJYAdSPMXJUBCuAujFGBPfBPjeduG2bJf/SgUjuc0sMEnEjZsm/VCCwVID0q0rciNlJ/qUCAe6zZgaBuBGzk7RHBQLcB/KvKnEj5geA4AJ7zSwF4kbM0z9jCv6lVQS5j8y/asSNmJ2T/EsH8gyChQxqfQAhrPxLA0KLOocibsS8wFn5lwbk0RMsoEjiSSeFmIF/qUAIc0oZiRsxA/9SgbCpaEFC3IgZ+JcK5DkEi0QB/qUDkdwnvYq7lrgRs+RfOpC7lK+/cQDBGh8V+eMVif/JKxcqiPiF+vqLoyb84qeq7CPtF4IIoeMvjqrKfQLClUOYvPhP5wAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}-2 & 0 & 0 & 0\\\\0 & 3 & 0 & 0\\\\0 & 0 & 5 & 0\\\\0 & 0 & 0 & 5\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡-2  0  0  0⎤\n",
       "⎢           ⎥\n",
       "⎢0   3  0  0⎥\n",
       "⎢           ⎥\n",
       "⎢0   0  5  0⎥\n",
       "⎢           ⎥\n",
       "⎣0   0  0  5⎦"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}3 & -2 & 4 & -2\\\\5 & 3 & -3 & -2\\\\5 & -2 & 2 & -2\\\\5 & -2 & -3 & 3\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡3  -2  4   -2⎤\n",
       "⎢             ⎥\n",
       "⎢5  3   -3  -2⎥\n",
       "⎢             ⎥\n",
       "⎢5  -2  2   -2⎥\n",
       "⎢             ⎥\n",
       "⎣5  -2  -3  3 ⎦"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P*D*P**-1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 96,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 96,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "P*D*P**-1==M"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 99,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\operatorname{PurePoly}{\\left( \\lambda^{4} - 11 \\lambda^{3} + 29 \\lambda^{2} + 35 \\lambda - 150, \\lambda, domain=\\mathbb{Z} \\right)}$$"
      ],
      "text/plain": [
       "PurePoly(lambda**4 - 11*lambda**3 + 29*lambda**2 + 35*lambda - 150, lambda, do\n",
       "main='ZZ')"
      ]
     },
     "execution_count": 99,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "M.charpoly()"
   ]
  }
 ],
 "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",
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   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.2"
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  "latex_envs": {
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   "current_citInitial": 1,
   "eqLabelWithNumbers": true,
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   "hotkeys": {
    "equation": "Ctrl-E",
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   "user_envs_cfg": false
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 "nbformat": 4,
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