{
 "metadata": {
  "name": "",
  "signature": "sha256:e9c2b606824fcdccfd75e24007928b297798ca40036f1a3916def3ce46fa44d2"
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 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Matrices Solutions"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from sympy import *\n",
      "init_printing()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 8
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use `row_del` and `row_insert` to go from one Matrix to the other."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matrix1(M):\n",
      "    \"\"\"\n",
      "    >>> M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])\n",
      "    >>> M\n",
      "    [1, 2, 3]\n",
      "    [4, 5, 6]\n",
      "    [7, 8, 9]\n",
      "    >>> matrix1(M)\n",
      "    [4, 5, 6]\n",
      "    [0, 0, 0]\n",
      "    [7, 8, 9]\n",
      "    \"\"\"\n",
      "    M.row_del(0)\n",
      "    M = M.row_insert(1, Matrix([[0, 0, 0]]))\n",
      "    return M"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M = Matrix([[1, 2, 3], [4, 5, 6], [7, 8, 9]])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}1 & 2 & 3\\\\4 & 5 & 6\\\\7 & 8 & 9\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRLvvzWYi3YlsmQtEVgAAA11JREFUWAntmduCmyAQhjlJt1EEyvu/a2eGQ0QZuslu\n90ovEkT9GH6B+UOETHQo8U1HyDwhZNIGjuWbuGJHmkoIlt/FfHL2E3i1z2uXktfarY9Lda2QMca1\n9rsH2zVNwB7fgkn10cqr315D6ZFMPj+CrVJqBlbUaIA+Do9ITaZwBUONmYG3DckqMWIsb4Mdtao5\nMEU6lOJfEdOTIsw6JVydDUeNPwWWKeYGBp8yNq54GezWAfFZZbbyBl4F6zlXiK0o9SLYVAmfMbaS\np/m7pp1qXgMvOAe8b6xjwSaK9S2wRK7Yx2CRHF4NKV9+JWIfYDGIukwtpHSHQQ0eddB0YK235HTW\nqHsmn5R1lgOLRcMiVVeSDjyAvV11g5t0txS3FE2BVvi5USEnK67brfCxrgYtuFawUcdYXNUl4jBJ\nERu6PT7l+QBQNfYVwmwTsNOqBtSiPBQCtuloVYZVu/dufp8ly4lKgNw7X3ACR/E+2G2H6E8RQ9aZ\ngtH3MgZLQGqSxuj6aruILTjFGRg9ruUclk0Bcw99kDAH403iT14edVV1PX72HrI0nsSBr1hwCM4i\nJkpk/D9EjNf34p9Ril8fv7HK0gidgFd6MnJa5PS/F+f956NJsYDvBue9qeLJsbHu2Aisi3HoLuFJ\nbvcYcffjZjJByK60GXABL6zGdOsETD8ylo0xQhAyaGm3Yku64QbvVKXEOxaptVIsVwitdBvmJ/Cl\nf29X3OAm3S3FLUVToBV+ZFTIAH4Dj9bsqWDhWktqp2twapWOqviVLpnuefcpsVme1k12jfIbJFJf\nF7+jFDFv27DcnVZEW6zOJeKVkqEqObMD072814nUJAvOqWmUTIn74BKTEEvCXaHI3GATxWpKSjxG\njGA7y/5rSrthX22NOKeQM3jy0qFZnVJgM4gmjdeSpU9gSwmRNBl87Fq6xO67WQexLnoMNsWEDqjg\nRXC4wZjMg2dwC2xqyrrhd4o4sGMNOLTtJnzZQhmAsWo8KmzK3mH4kC2mTTPDIj80Nt6PGViUScUZ\nAIMWyVcD1kuxTMGLQ3UXbryZAJdd7U4PlnP9Hgr2UKqzvsoVtW67vN0idL31CzV9xF8AnR+9wU2R\n/ywF5aTZbG6BfKbQ/mCx+IeIMewvw8/AjvfQHyzGiL/a/CifSWQDggAAAABJRU5ErkJggg==\n",
       "prompt_number": 11,
       "text": [
        "\u23a11  2  3\u23a4\n",
        "\u23a2       \u23a5\n",
        "\u23a24  5  6\u23a5\n",
        "\u23a2       \u23a5\n",
        "\u23a37  8  9\u23a6"
       ]
      }
     ],
     "prompt_number": 11
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix1(M)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}4 & 5 & 6\\\\0 & 0 & 0\\\\7 & 8 & 9\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
       "prompt_number": 12,
       "text": [
        "\u23a14  5  6\u23a4\n",
        "\u23a2       \u23a5\n",
        "\u23a20  0  0\u23a5\n",
        "\u23a2       \u23a5\n",
        "\u23a37  8  9\u23a6"
       ]
      }
     ],
     "prompt_number": 12
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Matrix Constructors"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Use the matrix constructors to construct the following matrices.  There may be more than one correct answer."
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\\left[\\begin{array}{ccc}4 & 0 & 0\\\\\\\\ \n",
      "0 & 4 & 0\\\\\\\\ \n",
      "0 & 0 & 4\\end{array}\\right]$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matrix2():\n",
      "    \"\"\"\n",
      "    >>> matrix2()\n",
      "    [4, 0, 0]\n",
      "    [0, 4, 0]\n",
      "    [0, 0, 4]\n",
      "    \"\"\"\n",
      "    return eye(3)*4\n",
      "    # OR return diag(4, 4, 4)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 13
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix2()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}4 & 0 & 0\\\\0 & 4 & 0\\\\0 & 0 & 4\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nZkkfWJyjL7LpKbg2v/C9AAAAAElFTkSuQmCC\n",
       "prompt_number": 7,
       "text": [
        "\u23a14  0  0\u23a4\n",
        "\u23a2       \u23a5\n",
        "\u23a20  4  0\u23a5\n",
        "\u23a2       \u23a5\n",
        "\u23a30  0  4\u23a6"
       ]
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\\left[\\begin{array}{}1 & 1 & 1 & 0\\\\\\\\1 & 1 & 1 & 0\\\\\\\\0 & 0 & 0 & 1\\end{array}\\right]$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matrix3():\n",
      "    \"\"\"\n",
      "    >>> matrix3()\n",
      "    [1, 1, 1, 0]\n",
      "    [1, 1, 1, 0]\n",
      "    [0, 0, 0, 1]\n",
      "    \"\"\"\n",
      "    return diag(ones(2, 3), 1)\n",
      "    # OR diag(ones(2, 3), ones(1, 1))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 14
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix3()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}1 & 1 & 1 & 0\\\\1 & 1 & 1 & 0\\\\0 & 0 & 0 & 1\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHgAAABLCAMAAACMVLPjAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nWi0qKQH4LwAAAABJRU5ErkJggg==\n",
       "prompt_number": 15,
       "text": [
        "\u23a11  1  1  0\u23a4\n",
        "\u23a2          \u23a5\n",
        "\u23a21  1  1  0\u23a5\n",
        "\u23a2          \u23a5\n",
        "\u23a30  0  0  1\u23a6"
       ]
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "$$\\left[\\begin{array}{}-1 & -1 & -1 & 0 & 0 & 0\\\\\\\\-1 & -1 & -1 & 0 & 0 & 0\\\\\\\\-1 & -1 & -1 & 0 & 0 & 0\\\\\\\\0 & 0 & 0 & 0 & 0 & 0\\\\\\\\0 & 0 & 0 & 0 & 0 & 0\\\\\\\\0 & 0 & 0 & 0 & 0 & 0\\end{array}\\right]$$"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matrix4():\n",
      "    \"\"\"\n",
      "    >>> matrix4()\n",
      "    [-1, -1, -1, 0, 0, 0]\n",
      "    [-1, -1, -1, 0, 0, 0]\n",
      "    [-1, -1, -1, 0, 0, 0]\n",
      "    [ 0,  0,  0, 0, 0, 0]\n",
      "    [ 0,  0,  0, 0, 0, 0]\n",
      "    [ 0,  0,  0, 0, 0, 0]\n",
      "    \"\"\"\n",
      "    return diag(-ones(3, 3), zeros(3, 3))\n",
      "    # OR diag(-ones(3, 3), 0, 0, 0)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 16
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix4()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}-1 & -1 & -1 & 0 & 0 & 0\\\\-1 & -1 & -1 & 0 & 0 & 0\\\\-1 & -1 & -1 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\\\0 & 0 & 0 & 0 & 0 & 0\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 17,
       "text": [
        "\u23a1-1  -1  -1  0  0  0\u23a4\n",
        "\u23a2                   \u23a5\n",
        "\u23a2-1  -1  -1  0  0  0\u23a5\n",
        "\u23a2                   \u23a5\n",
        "\u23a2-1  -1  -1  0  0  0\u23a5\n",
        "\u23a2                   \u23a5\n",
        "\u23a20   0   0   0  0  0\u23a5\n",
        "\u23a2                   \u23a5\n",
        "\u23a20   0   0   0  0  0\u23a5\n",
        "\u23a2                   \u23a5\n",
        "\u23a30   0   0   0  0  0\u23a6"
       ]
      }
     ],
     "prompt_number": 17
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": [
      "Advanced Methods"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Recall that if $f$ is an analytic function, then we can define $f(M)$ for any square matrix $M$ by \"plugging\" $M$ into the power series formula for $f(x)$. In other words, if $$f(x) = \\sum_{n=0}^\\infty a_n x^n,$$ then we define $f(M)$ by $$f(M) = \\sum_{n=0}^\\infty a_n M^n,$$ where $M^0$ is $I$, the identity matrix. "
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Furthermore, if $M$ is a diagonalizable matrix, that is, $M=PDP^{-1}$, where $D$ is diagonal, then $M^n = PD^nP^{-1}$ (because $M^n = \\left(PDP^{-1}\\right)\\left(PDP^{-1}\\right)\\cdots\\left(PDP^{-1}\\right)=PD\\left(P^{-1}P\\right)D\\left(P^{-1}P\\right)\\cdots DP^{-1} = PD^nP^{-1}$).\n",
      "\n",
      "But if \n",
      "\n",
      "$$ D = \\begin{bmatrix}\n",
      "         d_1 & 0 & \\cdots & 0 \\\\\\\\\n",
      "         0 & d_2 & \\cdots & 0 \\\\\\\\\n",
      "         \\vdots & \\vdots & \\ddots & \\vdots \\\\\\\\\n",
      "         0 & 0 & \\cdots & d_n\n",
      "      \\end{bmatrix}\n",
      "$$\n",
      "\n",
      "is a diagonal matrix, then \n",
      "\n",
      "$$ D^n = \\begin{bmatrix}\n",
      "         d_1^n & 0 & \\cdots & 0 \\\\\\\\\n",
      "         0 & d_2^n & \\cdots & 0 \\\\\\\\\n",
      "         \\vdots & \\vdots & \\ddots & \\vdots \\\\\\\\\n",
      "         0 & 0 & \\cdots & d_n^n\n",
      "      \\end{bmatrix}\n",
      "$$\n",
      "\n",
      "so that \n",
      "\n",
      "$$\n",
      "\\sum_{n=0}^\\infty a_n M^n = \\sum_{n=0}^\\infty a_n PD^nP^{-1} = P\\cdot\\begin{bmatrix}\n",
      "    \\sum_{n=0}^\\infty a_n d_1^n & 0 & \\cdots & 0 \\\\\\\\\n",
      "         0 & \\sum_{n=0}^\\infty a_n d_2^n & \\cdots & 0 \\\\\\\\\n",
      "         \\vdots & \\vdots & \\ddots & \\vdots \\\\\\\\\n",
      "         0 & 0 & \\cdots & \\sum_{n=0}^\\infty a_n d_n^n\n",
      "\\end{bmatrix}\\cdot P^{-1} = P\\cdot\\begin{bmatrix}\n",
      "    f(d_1) & 0 & \\cdots & 0 \\\\\\\\\n",
      "         0 & f(d_2) & \\cdots & 0 \\\\\\\\\n",
      "         \\vdots & \\vdots & \\ddots & \\vdots \\\\\\\\\n",
      "         0 & 0 & \\cdots & f(d_n)\n",
      "\\end{bmatrix}\\cdot P^{-1}\n",
      "$$"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Let's create some square matrices, which we will use throughout the exercises."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "x = symbols('x')\n",
      "A = Matrix([[1, 1], [1, 0]])\n",
      "M = Matrix([[3, 10, -30], [0, 3, 0], [0, 2, -3]])\n",
      "N = Matrix([[-1, -2, 0, 2], [-1, -1, 2, 1], [0, 0, 2, 0], [-1, -2, 2, 2]])"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 18
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "First, verify that these matrices are indeed diagonalizable."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "print(A.is_diagonalizable())\n",
      "print(M.is_diagonalizable())\n",
      "print(N.is_diagonalizable())"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "True\n",
        "True\n",
        "True\n"
       ]
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now, we want to write a function that computes $f(M)$, for diagonalizable matrix $M$ and analytic function $f$.\n",
      "\n",
      "However, there is one complication. We can use `diagonalize` to get `P` and `D`, but we need to apply the function to the diagonal of `D`. We might think that we could use `eigenvals` to get the eigenvalues of the matrix, since the diagonal values of `D` are just the eigenvalues of `M`, but the issue is that they could be in any order in `D`.\n",
      "\n",
      "Instead, we can use matrix slicing to get the diagonal values (or indeed, any value) of a matrix. There is not enough time in this tutorial (or room in this document) to discuss the full details of matrix slicing. For now, we just note that `M[i, j]` returns the element at position `i, j` (which is the `i + 1, j + 1`th element of the matrix, due to Python's 0-indexing). For example"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}3 & 10 & -30\\\\0 & 3 & 0\\\\0 & 2 & -3\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAH0AAABLCAMAAABqfXinAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nQmCC\n",
       "prompt_number": 20,
       "text": [
        "\u23a13  10  -30\u23a4\n",
        "\u23a2          \u23a5\n",
        "\u23a20  3    0 \u23a5\n",
        "\u23a2          \u23a5\n",
        "\u23a30  2   -3 \u23a6"
       ]
      }
     ],
     "prompt_number": 20
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M[0, 1]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$10$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAAPBAMAAAD0aukfAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAuxCrdpnvzWYy3UQi\nVIln4VgtAAAAaklEQVQIHWNgYBBiYGAKcZ3AwMCg+ImBgWsD00oGBmVjINOLgcEdKMoCZC5mYNh/\nAML8ysAw/wKYyfMPyHRAMB9gZUIUMIC0NUBMABp2/wCECbTiFQOEyXWBKZOBgTXsewQDU5wzyDlw\nAACa6iZBpbQ2vwAAAABJRU5ErkJggg==\n",
       "prompt_number": 21,
       "text": [
        "10"
       ]
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "That should be enough information to write the following function."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matrix_func(M, func):\n",
      "    \"\"\"\n",
      "    Computes M at func. Assumes that M is square diagonalizable.\n",
      "\n",
      "    >>> matrix_func(M, exp)\n",
      "    [exp(3), -5*exp(-3)/3 + 5*exp(3)/3, -5*exp(3) + 5*exp(-3)]\n",
      "    [     0,                    exp(3),                     0]\n",
      "    [     0,     -exp(-3)/3 + exp(3)/3,               exp(-3)]\n",
      "\n",
      "    Note that for the function exp, we can also just use M.exp()\n",
      "\n",
      "    >>> matrix_func(M, exp) == M.exp()\n",
      "    True\n",
      "\n",
      "    But for other functions, we have to do it this way.\n",
      "\n",
      "    >>> M.sin()\n",
      "    Traceback (most recent call last):\n",
      "    ...\n",
      "    AttributeError: Matrix has no attribute sin.\n",
      "    >>> matrix_func(N, sin)\n",
      "    [-sin(1), -2*sin(1),      0, 2*sin(1)]\n",
      "    [-sin(1),   -sin(1), sin(2),   sin(1)]\n",
      "    [      0,         0, sin(2),        0]\n",
      "    [-sin(1), -2*sin(1), sin(2), 2*sin(1)]\n",
      "    \n",
      "    Note that we could also use this to compute the series expansion of a matrix, \n",
      "    if we know the closed form of that expansion. For example, suppose we wanted to compute\n",
      "    \n",
      "    I + M + M**2 + M**3 + \u2026\n",
      "\n",
      "    The series\n",
      "\n",
      "    1 + x + x**2 + x**3 + \u2026\n",
      "\n",
      "    is equal to the function 1/(1 - x).\n",
      "\n",
      "    >>> matrix_func(M, Lambda(x, 1/(1 - x))) # Note, Lambda works just like lambda, but is symbolic\n",
      "    [-1/2, -5/4, 15/4]\n",
      "    [   0, -1/2,    0]\n",
      "    [   0, -1/4,  1/4]\n",
      "    \"\"\"\n",
      "    P, D = M.diagonalize()\n",
      "    diags = [func(D[i, i]) for i in range(M.shape[0])]\n",
      "    return P*diag(*diags)*P**-1"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 22
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func(M, exp)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}e^{3} & - \\frac{5}{3 e^{3}} + \\frac{5 e^{3}}{3} & - 5 e^{3} + \\frac{5}{e^{3}}\\\\0 & e^{3} & 0\\\\0 & - \\frac{1}{3 e^{3}} + \\frac{e^{3}}{3} & e^{-3}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQwAAABTBAMAAACYDSQ2AAAAMFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAMquZdlQQ3SJEie9m\nzbtvuCKIAAAGk0lEQVRoBe1aTYgcRRR+s7Pb2zP7N6AgeNkVg6inNSwii5Fht/0LKHMQPIg4ZCMJ\nWWFHQTR4SJ9EhLADCaigOCRBIRBILoKguIe9iAdXBYPgQBDMwYNsjIFdEcaq7n7Vr2peVfXuDDlI\n+rD96ns/9XVPd1e99xZgGMfZpfYwwgwa45dKa9AQw/CPq51hhJExggvnmrZYlcdeq3O6M+9uJnDp\nEqclGBscvaXdbK/3V2Jf3ihvEUdNrGQ2GigGn/bmUqhidU31bPDM+1Svtw2zC9GTienI/OR26tP/\nt/Jt3A8K5OMPU7gDO6xegWzwzHszWhI0asq2ar2kSlsZacJH6Si4HFqvAO2Z4Jk3gLh+QmPVMhlA\nZWG5ifHo+Uz0qHimIthcqlGYk5ng0js5NBpH/+DcEyyMx7Y45XvwMpxuHuFUCfZr8/1GquSCS+/k\n0GhA+VqKcn+DGxwKsFb7LXqKVwn0m94/qGODr6U3UacR3kIf8zxVg12CHUf5B5hpW/glJj89rH4s\nDB7E6AzSOxlQGuvt8F9lYQgTtZC+Ci+g+gqsxzsQ4LDvPIdIHnysjhhI72RAaRyLJ79QFoZQbY7O\nEUjR+AyW4HdYISpdVE55cEJDeicH0ggXl+NSdC5OUebvxUUKKhrV8x044ng2DkQPAejBCQ3pnRxI\n48U4nKfzeGRFw2MH12GtDXpwQkN5ZzTCXSirZ0kp7UJhGgAjl43gDhojX20+Y056+i15nATokWM7\ntUIaM0TVIwHCq4mzfA2qN4zgSAN9ZcjsbszMkxg2sSJ/Z3Ec6nbf6XZ/tJlRfLoF1V0avNztXv26\n251Do2y7hDTqiDNnXKOn2h+gFu9GMj5x+AHEzfNEC8o7M3UNxruRgNl2KaMxISx1Y+KJa3QIFxHV\naDwOhxA3z2WAqS0juEYj2y5lNEa3YCU2Y+BYrdH3biGk0bgfvkfcPIvX76XYCK7RyLZLGQ04H9XN\nEGSMa/QniGk0AOxbr7sOPghGcI0GVJJLQxoYnz+LNfpEVK/CTJzpdRrBl7wbjyoacoOQbZcK0RBr\ndPhzMLcK6xj4LAryHDzfoUOfXKlnFmKDgNulQjTEBmD6QNScjO6zTPGEBXfDcoOQbZeK0QhvzdRc\nIdcbLq1NRzYIRWjINVo8FU1LOPGtXt+w6Jww2SAUoSHX6OlGMG+JGbwN3zUtOidMNghFaMgNQCCe\nDYyJX1Uc331YvJT7OMgGoQgNcwb8qpr4AOP90FBf1QHmNVz3Q0Ms3smnzwg1yHB/NMRXdbhHTmMl\neqRoaC7zKeAbLCzXLGY5jZNwzGbU58tmPn1WJjDeDl43sWysaIy2QCzIBQ/MfAqaZ2ZPAzxn8VA0\nxuegQhMii72E88zHYcSo3hDlkJjBBaRoiA1jyZo66r555qPjvtFNUdJp8EaKxnoHSn/zNhTVMx+q\n8cqhSKpn67yZRkMl37ytRPXMx27HaBIaHUYhoD3SMDIfPqYF9dB4tSn9iv0oRuZjmZCHXT9K2IB7\nYukmH1FrWUHFpZmPAosK8hHd4I3HWlntS7ywY/4X1sh8+Jg2VLywr8S8Uj0b4vNVbfE2BDUyH6Ip\nIIrP1+cWM0UD3oRV/8fcyHwsQS3weCO4YlHlNFYu2Pbd1NWdVlHLfjlYXLJdaE6j3+02Indo0Jt9\n5278b+/GwA2/4TwbAzf8hkNj4IbfHmiUrtGHSpO9DT/NmhkUp1E66Fj50goWE78fYsuXxWmA2BPY\nDm/Djziy5cuh0MAKFpnMLrLly8FpFGz4EV5M+TKn4c1hLT+Kq+FH5s5FrnyZ0/DmsBYaroZfPncu\nseVLRcOfw+o0jmNgUs9DyHNmypeKhj+H1WmoOjGp57mnH/tTHNeFDVO+VDT8OayFBqnnuWmkWr58\nqWh406Xw0s06mUfdDVLPI2pDlKlvCvHlS41GgRxWhVc0FOISSOrLli9vDw1v6qvRMAsLqunHXOie\n7oY39VU0nDks7S9uC06k6Uf7hwzbFPKmvopGoRzW1vSzzp8pvKmvouHMYbFKzjf92KVbY+ZNfRUN\nZw6rquRs049dujUa3tQ3p+HKYfMqOdf0Y5dujYbR9NN1cpTT6NdRJK2S25p+zNJNnf1yURpp79HS\n9OOWbv/U1KIgjaz3yDf92KWbTuKXJQ38LziXddp7tFkwS7fNlMNPyf+COxpFz3JKDXP3HpmlW/P2\nDDajqO4xSdTO3iO/dBcJu1cbZ++RX7r3OkURe7P3qPuwS7du4h/9BxacEqsH+c5jAAAAAElFTkSu\nQmCC\n",
       "prompt_number": 23,
       "text": [
        "\u23a1         -3      3                \u23a4\n",
        "\u23a2 3    5\u22c5\u212f     5\u22c5\u212f        3      -3\u23a5\n",
        "\u23a2\u212f   - \u2500\u2500\u2500\u2500\u2500 + \u2500\u2500\u2500\u2500  - 5\u22c5\u212f  + 5\u22c5\u212f  \u23a5\n",
        "\u23a2        3      3                  \u23a5\n",
        "\u23a2                                  \u23a5\n",
        "\u23a2           3                      \u23a5\n",
        "\u23a20         \u212f               0       \u23a5\n",
        "\u23a2                                  \u23a5\n",
        "\u23a2         -3    3                  \u23a5\n",
        "\u23a2        \u212f     \u212f           -3      \u23a5\n",
        "\u23a20     - \u2500\u2500\u2500 + \u2500\u2500         \u212f        \u23a5\n",
        "\u23a3         3    3                   \u23a6"
       ]
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func(M, Lambda(x, 1/(1 - x)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}- \\frac{1}{2} & - \\frac{5}{4} & \\frac{15}{4}\\\\0 & - \\frac{1}{2} & 0\\\\0 & - \\frac{1}{4} & \\frac{1}{4}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIoAAABNCAMAAABOtAxaAAAASFBMVEX///8AAAAAAAAAAAAAAAAA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\nFUoAAAAASUVORK5CYII=\n",
       "prompt_number": 24,
       "text": [
        "\u23a1-1/2  -5/4  15/4\u23a4\n",
        "\u23a2                \u23a5\n",
        "\u23a2 0    -1/2   0  \u23a5\n",
        "\u23a2                \u23a5\n",
        "\u23a3 0    -1/4  1/4 \u23a6"
       ]
      }
     ],
     "prompt_number": 24
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Now lets investigate how this works in relation to the series expansion definition. Write a function that uses `matrix_func` and `series` to compute the approximation of a matrix evaluated at a function up to $O(M^n)$."
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "def matrix_func_series(M, func, n):\n",
      "    \"\"\"\n",
      "    Computes the approximation of the func(M) using the series definition up to O(M**n).\n",
      "\n",
      "    >>> matrix_func_series(M, exp, 10)\n",
      "    [22471/1120,  14953/448, -44859/448]\n",
      "    [         0, 22471/1120,          0]\n",
      "    [         0, 14953/2240,    83/2240]\n",
      "    >>> matrix_func_series(M, exp, 10).evalf()\n",
      "    [20.0633928571429, 33.3772321428571,  -100.131696428571]\n",
      "    [               0, 20.0633928571429,                  0]\n",
      "    [               0, 6.67544642857143, 0.0370535714285714]\n",
      "    >>> matrix_func(M, exp).evalf()\n",
      "    [20.0855369231877, 33.3929164246997,  -100.178749274099]\n",
      "    [               0, 20.0855369231877,                  0]\n",
      "    [               0, 6.67858328493993, 0.0497870683678639]\n",
      "\n",
      "    It's pretty close. Basically what we might expect for those values up to O(x**10).\n",
      "\n",
      "    >>> matrix_func_series(N, sin, 3)\n",
      "    [-1, -2, 0, 2]\n",
      "    [-1, -1, 2, 1]\n",
      "    [ 0,  0, 2, 0]\n",
      "    [-1, -2, 2, 2]\n",
      "    >>> matrix_func(N, sin).evalf()\n",
      "    [-0.841470984807897,  -1.68294196961579,                 0,  1.68294196961579]\n",
      "    [-0.841470984807897, -0.841470984807897, 0.909297426825682, 0.841470984807897]\n",
      "    [                 0,                  0, 0.909297426825682,                 0]\n",
      "    [-0.841470984807897,  -1.68294196961579, 0.909297426825682,  1.68294196961579]\n",
      "\n",
      "    It's not as close, because we used O(x**3), but clearly still the same thing.\n",
      "\n",
      "    >>> matrix_func_series(M, Lambda(x, 1/(1 - x)), 10)\n",
      "    [29524, 73810, -221430]\n",
      "    [    0, 29524,       0]\n",
      "    [    0, 14762,  -14762]\n",
      "    >>> matrix_func(M, Lambda(x, 1/(1 - x)))\n",
      "    [-1/2, -5/4, 15/4]\n",
      "    [   0, -1/2,    0]\n",
      "    [   0, -1/4,  1/4]\n",
      "\n",
      "    Woah! That one's not close at all. What is happening here?  Let's try more terms\n",
      "\n",
      "    >>> matrix_func_series(M, Lambda(x, 1/(1 - x)), 100)\n",
      "    [257688760366005665518230564882810636351053761000, 644221900915014163795576412207026590877634402500, -1932665702745042491386729236621079772632903207500]\n",
      "    [                                               0, 257688760366005665518230564882810636351053761000,                                                  0]\n",
      "    [                                               0, 128844380183002832759115282441405318175526880500,  -128844380183002832759115282441405318175526880500]\n",
      "    \n",
      "    It just keeps getting bigger. In fact, the series diverges. Recall that \n",
      "    1/(1 - x) = 1 + x + x**2 + x**3 + \u2026 *only if* |x| < 1.  But the eigenvalues \n",
      "    of M are bigger than 1 in absolute value.\n",
      "    \n",
      "    >>> M.eigenvals()\n",
      "    {3: 2, -3: 1}\n",
      "\n",
      "    In fact, 1/(1 - M) is mathematically defined via the analytic continuation \n",
      "    of the series expansion 1 + x + x**2 + \u2026, which is just 1/(1 - x).  This is\n",
      "    well-defined as long as none of the eigenvalues of M are equal to 1.  Let's\n",
      "    try it on N.\n",
      "\n",
      "    >>> matrix_func(N, Lambda(x, 1/(1 - x)))\n",
      "    [nan, -oo, nan,  oo]\n",
      "    [nan, nan, nan, nan]\n",
      "    [nan, nan, nan, nan]\n",
      "    [nan, -oo, nan,  oo]\n",
      "    \n",
      "    That didn't work. What are the eigenvalues of N?\n",
      "\n",
      "    >>> N.eigenvals()\n",
      "    {1: 1, 2: 1, -1: 1, 0: 1}\n",
      "\n",
      "    Ah, the first one is 1, so we cannot define 1/(1 - N). \n",
      "    \"\"\"\n",
      "    x = Dummy('x') # This works even if func already contains Symbol('x')\n",
      "    series_func = Lambda(x, func(x).series(x, 0, n).removeO())\n",
      "    return matrix_func(M, series_func)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 26
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func_series(M, exp, 10)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{22471}{1120} & \\frac{14953}{448} & - \\frac{44859}{448}\\\\0 & \\frac{22471}{1120} & 0\\\\0 & \\frac{14953}{2240} & \\frac{83}{2240}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANQAAABNCAMAAADpabqhAAAAaVBMVEX///8AAAAAAAAAAAAAAAAA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\ngg==\n",
       "prompt_number": 27,
       "text": [
        "\u23a122471  14953  -44859 \u23a4\n",
        "\u23a2\u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500  \u2500\u2500\u2500\u2500\u2500\u2500\u2500\u23a5\n",
        "\u23a2 1120   448     448  \u23a5\n",
        "\u23a2                     \u23a5\n",
        "\u23a2       22471         \u23a5\n",
        "\u23a2  0    \u2500\u2500\u2500\u2500\u2500     0   \u23a5\n",
        "\u23a2        1120         \u23a5\n",
        "\u23a2                     \u23a5\n",
        "\u23a2       14953    83   \u23a5\n",
        "\u23a2  0    \u2500\u2500\u2500\u2500\u2500   \u2500\u2500\u2500\u2500  \u23a5\n",
        "\u23a3        2240   2240  \u23a6"
       ]
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func_series(M, exp, 10).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}20.0633928571429 & 33.3772321428571 & -100.131696428571\\\\0 & 20.0633928571429 & 0\\\\0 & 6.67544642857143 & 0.0370535714285714\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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       "prompt_number": 28,
       "text": [
        "\u23a120.0633928571429  33.3772321428571  -100.131696428571 \u23a4\n",
        "\u23a2                                                      \u23a5\n",
        "\u23a2       0          20.0633928571429          0         \u23a5\n",
        "\u23a2                                                      \u23a5\n",
        "\u23a3       0          6.67544642857143  0.0370535714285714\u23a6"
       ]
      }
     ],
     "prompt_number": 28
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func(M, exp).evalf()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}20.0855369231877 & 33.3929164246997 & -100.178749274099\\\\0 & 20.0855369231877 & 0\\\\0 & 6.67858328493993 & 0.0497870683678639\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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qiWNFXYRMF78XlpGwo2v+QsHSZnhOmhfW4OkPGCwo1dRY+V0+ZKKcPvjDXPeZDz4c0Scs\nkInA6BoWyHTBER0UyERgdA0LZLrgiA4Umax/qxWZfeiG+BJxUu6bn//ch+6TD/rAx+fna/8F5VBN\n7F3LwaQAAAAASUVORK5CYII=\n",
       "prompt_number": 29,
       "text": [
        "\u23a120.0855369231877  33.3929164246997  -100.178749274099 \u23a4\n",
        "\u23a2                                                      \u23a5\n",
        "\u23a2       0          20.0855369231877          0         \u23a5\n",
        "\u23a2                                                      \u23a5\n",
        "\u23a3       0          6.67858328493993  0.0497870683678639\u23a6"
       ]
      }
     ],
     "prompt_number": 29
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func_series(N, sin, 3)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}-1 & -2 & 0 & 2\\\\-1 & -1 & 2 & 1\\\\0 & 0 & 2 & 0\\\\-1 & -2 & 2 & 2\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAJgAAABkCAMAAABNTAlxAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\n056Dn+vnJ1KdM/8A0rpFHAD89/AAAAAASUVORK5CYII=\n",
       "prompt_number": 30,
       "text": [
        "\u23a1-1  -2  0  2\u23a4\n",
        "\u23a2            \u23a5\n",
        "\u23a2-1  -1  2  1\u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a20   0   2  0\u23a5\n",
        "\u23a2            \u23a5\n",
        "\u23a3-1  -2  2  2\u23a6"
       ]
      }
     ],
     "prompt_number": 30
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func_series(M, Lambda(x, 1/(1 - x)), 100)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}257688760366005665518230564882810636351053761000 & 644221900915014163795576412207026590877634402500 & -1932665702745042491386729236621079772632903207500\\\\0 & 257688760366005665518230564882810636351053761000 & 0\\\\0 & 128844380183002832759115282441405318175526880500 & -128844380183002832759115282441405318175526880500\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": 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j2KeCt2Rb1UyG+yen4j7fcV2buYYy2Q9Ds6yFcNlMSJrhiffU13QqngkRNTzc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\nhpAAAAAASUVORK5CYII=\n",
       "prompt_number": 31,
       "text": [
        "\u23a1257688760366005665518230564882810636351053761000  644221900915014163795576412\n",
        "\u23a2                                                                             \n",
        "\u23a2                       0                          257688760366005665518230564\n",
        "\u23a2                                                                             \n",
        "\u23a3                       0                          128844380183002832759115282\n",
        "\n",
        "207026590877634402500  -1932665702745042491386729236621079772632903207500\u23a4\n",
        "                                                                         \u23a5\n",
        "882810636351053761000                          0                         \u23a5\n",
        "                                                                         \u23a5\n",
        "441405318175526880500  -128844380183002832759115282441405318175526880500 \u23a6"
       ]
      }
     ],
     "prompt_number": 31
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "M.eigenvals()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}-3 : 1, & 3 : 2\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAIsAAAAaBAMAAAB1DbIqAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
       "prompt_number": 32,
       "text": [
        "{-3: 1, 3: 2}"
       ]
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "matrix_func(N, Lambda(x, 1/(1 - x)))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\mathrm{NaN} & \\tilde{\\infty} & \\mathrm{NaN} & \\tilde{\\infty}\\\\\\mathrm{NaN} & \\mathrm{NaN} & \\mathrm{NaN} & \\mathrm{NaN}\\\\\\mathrm{NaN} & \\mathrm{NaN} & \\mathrm{NaN} & \\mathrm{NaN}\\\\\\mathrm{NaN} & \\tilde{\\infty} & \\mathrm{NaN} & \\tilde{\\infty}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAPQAAABkCAMAAACGl+2PAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\nfe0LQGRyr2v2F9EolM42ZqoDAAAAAElFTkSuQmCC\n",
       "prompt_number": 33,
       "text": [
        "\u23a1nan  zoo  nan  zoo\u23a4\n",
        "\u23a2                  \u23a5\n",
        "\u23a2nan  nan  nan  nan\u23a5\n",
        "\u23a2                  \u23a5\n",
        "\u23a2nan  nan  nan  nan\u23a5\n",
        "\u23a2                  \u23a5\n",
        "\u23a3nan  zoo  nan  zoo\u23a6"
       ]
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "N.eigenvals()"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\begin{Bmatrix}-1 : 1, & 0 : 1, & 1 : 1, & 2 : 1\\end{Bmatrix}$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAQwAAAAaBAMAAACwWn+SAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nN5YrFYF3pa30z/T9BlpbuVQBNRoeAAAAAElFTkSuQmCC\n",
       "prompt_number": 34,
       "text": [
        "{-1: 1, 0: 1, 1: 1, 2: 1}"
       ]
      }
     ],
     "prompt_number": 34
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [],
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}