{
 "metadata": {
  "language": "Julia",
  "name": "18.06 Sep 9 2013 Lecture 3"
 },
 "nbformat": 3,
 "nbformat_minor": 0,
 "worksheets": [
  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "Matrix Multiply and Inverses"
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Many ways to multiply matrices"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A=rand(1:3,4,6)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 4,
       "text": "4x6 Array{Int64,2}:\n 3  3  1  3  3  1\n 2  2  2  3  1  3\n 3  1  2  2  1  3\n 3  3  3  2  1  2"
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B=rand(1:3,7,8)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 5,
       "text": "7x8 Array{Int64,2}:\n 3  3  3  3  3  1  1  2\n 1  1  1  1  3  2  3  1\n 3  2  3  2  2  3  2  2\n 3  3  1  2  3  2  3  1\n 3  3  3  1  3  2  3  2\n 2  1  3  2  2  1  2  3\n 2  3  1  1  1  2  1  2"
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*B # matmul only works if A is m x n  and B is n x p",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "LoadError",
       "evalue": "*: argument shapes do not match\nat In[6]:1",
       "output_type": "pyerr",
       "traceback": [
        "*: argument shapes do not match\nat In[6]:1",
        " in generic_matmatmul at linalg/matmul.jl:409",
        " in generic_matmatmul at linalg/matmul.jl:398",
        " in * at linalg/matmul.jl:391"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B=rand(1:3,6,8)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 7,
       "text": "6x8 Array{Int64,2}:\n 2  2  2  2  2  2  3  2\n 3  3  2  2  1  1  3  2\n 1  1  2  1  2  1  2  1\n 3  2  2  2  1  1  2  2\n 1  1  1  1  2  2  2  2\n 1  2  2  1  3  1  3  2"
      }
     ],
     "prompt_number": 7
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*B",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 8,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 8
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[3,:]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 9,
       "text": "1x6 Array{Int64,2}:\n 3  1  2  2  1  3"
      }
     ],
     "prompt_number": 9
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B[:,4]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 10,
       "text": "6-element Array{Int64,1}:\n 2\n 2\n 1\n 2\n 1\n 1"
      }
     ],
     "prompt_number": 10
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[3,:]*B[:,4]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 13,
       "text": "1-element Array{Int64,1}:\n 18"
      }
     ],
     "prompt_number": 13
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Linear Combinations of Columns of A"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[A*B[:,1] A*B[:,2] A*B[:,3] A*B[:,4] A*B[:,5] A*B[:,6]  A*B[:,7] A*B[:,8]]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 19,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 19
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*B  # Every column of A*B is some linear combination of columns of A",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 15,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 15
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Linear Combinations of Rows of B"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[ A[1,:]*B; A[2,:]*B; A[3,:]*B; A[4,:]*B]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 23,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 23
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[1,:] % 3 x first row of B + 3 * 2nd row + 1* 3rd row + etc.",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 21,
       "text": "1x6 Array{Int64,2}:\n 3  3  1  3  3  1"
      }
     ],
     "prompt_number": 21
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Outer Product Matrix Multiply"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "v=rand(5,1); w=rand(1,5)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 25,
       "text": "1x5 Array{Float64,2}:\n 0.648415  0.107518  0.49727  0.0484907  0.148389"
      }
     ],
     "prompt_number": 25
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "v*w;w*v",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 27,
       "text": "1x1 Array{Float64,2}:\n 0.341224"
      }
     ],
     "prompt_number": 27
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "[1:12]*[1:12]'  # Outer Products (recognize the 3rd grade mult table)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 32,
       "text": "12x12 Array{Int64,2}:\n  1   2   3   4   5   6   7   8    9   10   11   12\n  2   4   6   8  10  12  14  16   18   20   22   24\n  3   6   9  12  15  18  21  24   27   30   33   36\n  4   8  12  16  20  24  28  32   36   40   44   48\n  5  10  15  20  25  30  35  40   45   50   55   60\n  6  12  18  24  30  36  42  48   54   60   66   72\n  7  14  21  28  35  42  49  56   63   70   77   84\n  8  16  24  32  40  48  56  64   72   80   88   96\n  9  18  27  36  45  54  63  72   81   90   99  108\n 10  20  30  40  50  60  70  80   90  100  110  120\n 11  22  33  44  55  66  77  88   99  110  121  132\n 12  24  36  48  60  72  84  96  108  120  132  144"
      }
     ],
     "prompt_number": 32
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*B",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 33,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 33
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[:,1]*B[1,:] + A[:,2]*B[2,:] + A[:,3]*B[3,:] + A[:,4]*B[4,:]+A[:,5]*B[5,:]+A[:,6]*B[6,:]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 47,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 47
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*B\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 43,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 43
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 44,
       "text": "4x6 Array{Int64,2}:\n 3  3  1  3  3  1\n 2  2  2  3  1  3\n 3  1  2  2  1  3\n 3  3  3  2  1  2"
      }
     ],
     "prompt_number": 44
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 45,
       "text": "6x8 Array{Int64,2}:\n 2  2  2  2  2  2  3  2\n 3  3  2  2  1  1  3  2\n 1  1  2  1  2  1  2  1\n 3  2  2  2  1  1  2  2\n 1  1  1  1  2  2  2  2\n 1  2  2  1  3  1  3  2"
      }
     ],
     "prompt_number": 45
    },
    {
     "cell_type": "heading",
     "level": 2,
     "metadata": {},
     "source": "Block Matrix Multiply"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[1:2,1:2]*B[1:2,1:2] + A[1:2,3:4]*B[3:4,1:2]+ A[1:2,5:6]*B[5:6,1:2]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 49,
       "text": "2x2 Array{Int64,2}:\n 29  27\n 25  25"
      }
     ],
     "prompt_number": 49
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*B\n",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 50,
       "text": "4x8 Array{Int64,2}:\n 29  27  25  23  23  20  35  27\n 25  25  25  20  24  16  33  24\n 21  22  23  18  24  16  31  22\n 27  27  27  22  25  18  36  25"
      }
     ],
     "prompt_number": 50
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[3:4,1:2]*B[1:2,1:2] + A[3:4,3:4]*B[3:4,1:2]+ A[3:4,5:6]*B[5:6,1:2]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 51,
       "text": "2x2 Array{Int64,2}:\n 21  22\n 27  27"
      }
     ],
     "prompt_number": 51
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A[3:4,1:2]*B[1:2,5:6] + A[3:4,3:4]*B[3:4,5:6]+ A[3:4,5:6]*B[5:6,5:6\n                                                            ]",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 52,
       "text": "2x2 Array{Int64,2}:\n 24  16\n 25  18"
      }
     ],
     "prompt_number": 52
    },
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": "Matrix Inverses"
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A= rand(3,4)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 53,
       "text": "3x4 Array{Float64,2}:\n 0.973557  0.278488  0.300054  0.723139\n 0.222204  0.860992  0.562578  0.846122\n 0.130402  0.852375  0.108227  0.26576 "
      }
     ],
     "prompt_number": 53
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "eye(4)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 54,
       "text": "4x4 Array{Float64,2}:\n 1.0  0.0  0.0  0.0\n 0.0  1.0  0.0  0.0\n 0.0  0.0  1.0  0.0\n 0.0  0.0  0.0  1.0"
      }
     ],
     "prompt_number": 54
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A=rand(1:3,3,3)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 72,
       "text": "3x3 Array{Int64,2}:\n 1  1  2\n 1  1  2\n 1  1  1"
      }
     ],
     "prompt_number": 72
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "inv(A)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "ename": "LoadError",
       "evalue": "SingularException(2)\nat In[73]:1",
       "output_type": "pyerr",
       "traceback": [
        "SingularException(2)\nat In[73]:1",
        " in inv at linalg/factorization.jl:231",
        " in inv at linalg/dense.jl:425"
       ]
      }
     ],
     "prompt_number": 73
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B=ans ",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 59,
       "text": "3x3 Array{Float64,2}:\n -0.0  -0.333333   0.666667\n -1.0   1.66667   -0.333333\n  1.0  -1.0        0.0     "
      }
     ],
     "prompt_number": 59
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "A*B",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 60,
       "text": "3x3 Array{Float64,2}:\n 1.0  -4.44089e-16  1.11022e-16\n 0.0   1.0          1.11022e-16\n 0.0  -1.11022e-16  1.0        "
      }
     ],
     "prompt_number": 60
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "B*A",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 61,
       "text": "3x3 Array{Float64,2}:\n 1.0  0.0   0.0        \n 0.0  1.0  -2.22045e-16\n 0.0  0.0   1.0        "
      }
     ],
     "prompt_number": 61
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "2.0^(-52)",
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "pyout",
       "prompt_number": 63,
       "text": "2.220446049250313e-16"
      }
     ],
     "prompt_number": 63
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": "",
     "language": "python",
     "metadata": {},
     "outputs": []
    }
   ],
   "metadata": {}
  }
 ]
}