tobydriscoll · August 30, 2016 21:25
diff --git a/TB-Lecture-02-julia b/TB-Lecture-02-julia
 {
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 2: Orthogonal vectors and matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "With real vectors and matrices, the transpose operation is simple and familiar. It also happens to correspond to what we call the **adjoint** mathematically. In the complex case, one also has to conjugate the entries to keep the mathematical structure intact. We call this operator the **hermitian** of a matrix and use a star superscript for it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2x4 Array{Complex{Float64},2}:\n",
       " 0.840791+0.839765im  0.505249+0.883819im  …   0.583826+0.634644im\n",
       " 0.971119+0.123017im  0.119297+0.765509im     0.0494804+0.222855im"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(2,4) + 1im*rand(2,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4x2 Array{Complex{Float64},2}:\n",
       "  0.840791-0.839765im   0.971119-0.123017im\n",
       "  0.505249-0.883819im   0.119297-0.765509im\n",
       " 0.0546813-0.662234im   0.715531-0.348226im\n",
       "  0.583826-0.634644im  0.0494804-0.222855im"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Aadjoint = A'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To get plain transpose, use a `.^` operator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4x2 Array{Complex{Float64},2}:\n",
       "  0.840791+0.839765im   0.971119+0.123017im\n",
       "  0.505249+0.883819im   0.119297+0.765509im\n",
       " 0.0546813+0.662234im   0.715531+0.348226im\n",
       "  0.583826+0.634644im  0.0494804+0.222855im"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Atrans = A.'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Inner products"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If **u** and **v** are column vectors of the same length, then their **inner product** is $\\mathbf{u}^*\\mathbf{v}$. The result is a scalar. (In Julia, though, the result is a 1-by-1 matrix.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{Complex{Int64},1}:\n",
       " -2-3im"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = [ 4; -1; 2+2im ]\n",
    "v = [ -1; 1im; 1 ]\n",
    "innerprod = u'*v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inner product has geometric significance. It is used to define length through the 2-norm,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{Complex{Int64},1}:\n",
       " 25+0im"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "length_u_squared = u'*u"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "25.0"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum( abs(u).^2 )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.0"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "norm_u = norm(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It also defines the angle between two vectors as a generalization of the familiar dot product."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{Complex{Float64},1}:\n",
       " -0.23094-0.34641im"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cos_theta = (u'*v) / ( norm(u)*norm(v) )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The angle may be complex when the vectors are complex! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{Complex{Float64},1}:\n",
       " 1.79019+0.34786im"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "theta = acos(cos_theta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The operations of inverse and hermitian commute."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4x4 Array{Complex{Float64},2}:\n",
       " 0.0886673+0.982318im  -0.490957-1.13907im   …     1.2432-0.145164im\n",
       "  -1.07021-1.818im       1.13836+1.15101im      -0.139613+0.474516im\n",
       " -0.436714+0.051986im   0.352478+0.521735im      0.793142+0.407391im\n",
       "  0.882468+1.38138im   -0.467233-0.833646im     -0.954656-0.650509im"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(4,4)+1im*rand(4,4);  (inv(A))'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4x4 Array{Complex{Float64},2}:\n",
       " 0.0886673+0.982318im  -0.490957-1.13907im   …     1.2432-0.145164im\n",
       "  -1.07021-1.818im       1.13836+1.15101im      -0.139613+0.474516im\n",
       " -0.436714+0.051986im   0.352478+0.521735im      0.793142+0.407391im\n",
       "  0.882468+1.38138im   -0.467233-0.833646im     -0.954656-0.650509im"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "inv(A')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So we just write $\\mathbf{A}^{-*}$ for either case. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Orthogonality"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Orthogonality, which is the multidimensional extension of perpendicularity, means that $\\cos \\theta=0$, i.e., that the inner product between vectors is zero. A collection of vectors is orthogonal if they are all pairwise orthogonal. \n",
    "\n",
    "Don't worry about how we are creating the vectors here for now."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5x3 Array{Float64,2}:\n",
       " -0.103283  -0.0246192   0.792393\n",
       " -0.33304   -0.475506   -0.348076\n",
       " -0.629735   0.364999    0.291983\n",
       " -0.13881   -0.797678    0.295044\n",
       " -0.680135   0.061426   -0.28045 "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q = full( qrfact(rand(5,3))[:Q] )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since $\\mathbf{Q}^*\\mathbf{Q}$ is a matrix of all inner products between columns of $\\mathbf{Q}$, those columns are orthogonal if and only if that matrix is diagonal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3x3 Array{Float64,2}:\n",
       "  1.0          -6.40858e-17  4.16557e-17\n",
       " -6.40858e-17   1.0          7.12044e-17\n",
       "  4.16557e-17   7.12044e-17  1.0        "
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "QhQ = Q'*Q"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In fact we have a stronger condition here: the columns are **orthonormal**, meaning that they are orthogonal and each has 2-norm equal to 1. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given any other vector of length 5, we can compute its inner product with each of the columns of $\\mathbf{Q}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3x1 Array{Float64,2}:\n",
       " -1.11399   \n",
       " -0.901804  \n",
       "  0.00955111"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u = rand(5,1);  c = Q'*u"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can then use these coefficients to find a vector in the column space of $\\mathbf{Q}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5x1 Array{Float64,2}:\n",
       " 0.144826\n",
       " 0.796491\n",
       " 0.375148\n",
       " 0.8768  \n",
       " 0.699589"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v = Q*c"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As explained in the text, $\\mathbf{r} = \\mathbf{u}-\\mathbf{v}$ is orthogonal to all of the columns of $\\mathbf{Q}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3x1 Array{Float64,2}:\n",
       " -2.24722e-16\n",
       "  7.41632e-17\n",
       "  1.13865e-16"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r = u-v;  Q'*r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consequently, we have decomposed $\\mathbf{u}=\\mathbf{v}+\\mathbf{r}$ into the sum of two orthogonal parts, one lying in the range of $\\mathbf{Q}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1x1 Array{Float64,2}:\n",
       " 1.90887e-16"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "v'*r"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Unitary matrices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We just saw that a matrix whose columns are orthonormal is pretty special. It becomes even more special if the matrix is also square, in which case we call it **unitary**. (In the real case, such matrices are confusingly called _orthogonal_. Ugh.) Say $\\mathbf{Q}$ is unitary and $m\\times m$. Then $\\mathbf{Q}^*\\mathbf{Q}$ is an $m\\times m$ identity matrix---that is, $\\mathbf{Q}^*=\\mathbf{Q}^{-1}$! It can't get much easier in terms of finding the inverse of a matrix. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5x5 Array{Float64,2}:\n",
       " 2.77556e-16  5.55112e-17  1.57009e-16  3.10317e-16  1.75542e-16\n",
       " 1.24127e-16  2.48253e-16  3.04364e-16  1.41271e-16  1.24127e-16\n",
       " 3.88578e-16  2.77556e-16  3.60822e-16  1.57009e-16  1.49468e-16\n",
       " 1.77858e-16  1.19696e-16  2.22045e-16  1.24127e-16  8.32667e-17\n",
       " 2.48253e-16  1.75542e-16  1.57009e-16  2.65135e-16  1.19834e-16"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q = full( qrfact(rand(5,5)+1im*rand(5,5))[:Q] );\n",
    "abs( inv(Q) - Q' )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The rank of $\\mathbf{Q}$ is $m$, so continuing the discussion above, the original vector $\\mathbf{u}$ lies in its column space. Hence the remainder $\\mathbf{r}=\\boldsymbol{0}$. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5x1 Array{Complex{Float64},2}:\n",
       " -1.94289e-16-7.63278e-17im\n",
       "          0.0+1.38778e-16im\n",
       " -2.77556e-16+2.35922e-16im\n",
       " -2.22045e-16-3.33067e-16im\n",
       " -2.22045e-16+1.11022e-16im"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c = Q'*u; \n",
    "v = Q*c;\n",
    "r = u - v"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is another way to arrive at a fact we already knew: Multiplication by $\\mathbf{Q}^*=\\mathbf{Q}^{-1}$ changes the basis to the columns of $\\mathbf{Q}$."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Julia 0.4.3",
   "language": "julia",
   "name": "julia-0.4"
  },
  "language_info": {
   "file_extension": ".jl",
   "mimetype": "application/julia",
   "name": "julia",
   "version": "0.4.3"
  }
 },
 "nbformat": 4,
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 }
	{
	"cells": [
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"# Lecture 2: Orthogonal vectors and matrices"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {
	"collapsed": true
	},
	"source": [
	"With real vectors and matrices, the transpose operation is simple and familiar. It also happens to correspond to what we call the adjoint mathematically. In the complex case, one also has to conjugate the entries to keep the mathematical structure intact. We call this operator the hermitian of a matrix and use a star superscript for it."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 1,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"2x4 Array{Complex{Float64},2}:\n",
	" 0.840791+0.839765im 0.505249+0.883819im … 0.583826+0.634644im\n",
	" 0.971119+0.123017im 0.119297+0.765509im 0.0494804+0.222855im"
	]
	},
	"execution_count": 1,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"A = rand(2,4) + 1im*rand(2,4)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 2,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"4x2 Array{Complex{Float64},2}:\n",
	" 0.840791-0.839765im 0.971119-0.123017im\n",
	" 0.505249-0.883819im 0.119297-0.765509im\n",
	" 0.0546813-0.662234im 0.715531-0.348226im\n",
	" 0.583826-0.634644im 0.0494804-0.222855im"
	]
	},
	"execution_count": 2,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Aadjoint = A'"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"To get plain transpose, use a `.^` operator."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 3,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"4x2 Array{Complex{Float64},2}:\n",
	" 0.840791+0.839765im 0.971119+0.123017im\n",
	" 0.505249+0.883819im 0.119297+0.765509im\n",
	" 0.0546813+0.662234im 0.715531+0.348226im\n",
	" 0.583826+0.634644im 0.0494804+0.222855im"
	]
	},
	"execution_count": 3,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Atrans = A.'"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"## Inner products"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"If u and v are column vectors of the same length, then their inner product is $\\mathbf{u}^*\\mathbf{v}$. The result is a scalar. (In Julia, though, the result is a 1-by-1 matrix.)"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 4,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"1-element Array{Complex{Int64},1}:\n",
	" -2-3im"
	]
	},
	"execution_count": 4,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"u = [ 4; -1; 2+2im ]\n",
	"v = [ -1; 1im; 1 ]\n",
	"innerprod = u'*v"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"The inner product has geometric significance. It is used to define length through the 2-norm,"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 5,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"1-element Array{Complex{Int64},1}:\n",
	" 25+0im"
	]
	},
	"execution_count": 5,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"length_u_squared = u'*u"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 6,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"25.0"
	]
	},
	"execution_count": 6,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"sum( abs(u).^2 )"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 7,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"5.0"
	]
	},
	"execution_count": 7,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"norm_u = norm(u)"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"It also defines the angle between two vectors as a generalization of the familiar dot product."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 8,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"1-element Array{Complex{Float64},1}:\n",
	" -0.23094-0.34641im"
	]
	},
	"execution_count": 8,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"cos_theta = (u'v) / ( norm(u)norm(v) )"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"The angle may be complex when the vectors are complex! "
	]
	},
	{
	"cell_type": "code",
	"execution_count": 9,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"1-element Array{Complex{Float64},1}:\n",
	" 1.79019+0.34786im"
	]
	},
	"execution_count": 9,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"theta = acos(cos_theta)"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"The operations of inverse and hermitian commute."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 10,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"4x4 Array{Complex{Float64},2}:\n",
	" 0.0886673+0.982318im -0.490957-1.13907im … 1.2432-0.145164im\n",
	" -1.07021-1.818im 1.13836+1.15101im -0.139613+0.474516im\n",
	" -0.436714+0.051986im 0.352478+0.521735im 0.793142+0.407391im\n",
	" 0.882468+1.38138im -0.467233-0.833646im -0.954656-0.650509im"
	]
	},
	"execution_count": 10,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"A = rand(4,4)+1im*rand(4,4); (inv(A))'"
	]
	},
	{
	"cell_type": "code",
	"execution_count": 11,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"4x4 Array{Complex{Float64},2}:\n",
	" 0.0886673+0.982318im -0.490957-1.13907im … 1.2432-0.145164im\n",
	" -1.07021-1.818im 1.13836+1.15101im -0.139613+0.474516im\n",
	" -0.436714+0.051986im 0.352478+0.521735im 0.793142+0.407391im\n",
	" 0.882468+1.38138im -0.467233-0.833646im -0.954656-0.650509im"
	]
	},
	"execution_count": 11,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"inv(A')"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"So we just write $\\mathbf{A}^{-*}$ for either case. "
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"## Orthogonality"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Orthogonality, which is the multidimensional extension of perpendicularity, means that $\\cos \\theta=0$, i.e., that the inner product between vectors is zero. A collection of vectors is orthogonal if they are all pairwise orthogonal. \n",
	"\n",
	"Don't worry about how we are creating the vectors here for now."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 12,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"5x3 Array{Float64,2}:\n",
	" -0.103283 -0.0246192 0.792393\n",
	" -0.33304 -0.475506 -0.348076\n",
	" -0.629735 0.364999 0.291983\n",
	" -0.13881 -0.797678 0.295044\n",
	" -0.680135 0.061426 -0.28045 "
	]
	},
	"execution_count": 12,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Q = full( qrfact(rand(5,3))[:Q] )"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Since $\\mathbf{Q}^*\\mathbf{Q}$ is a matrix of all inner products between columns of $\\mathbf{Q}$, those columns are orthogonal if and only if that matrix is diagonal."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 13,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"3x3 Array{Float64,2}:\n",
	" 1.0 -6.40858e-17 4.16557e-17\n",
	" -6.40858e-17 1.0 7.12044e-17\n",
	" 4.16557e-17 7.12044e-17 1.0 "
	]
	},
	"execution_count": 13,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"QhQ = Q'*Q"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"In fact we have a stronger condition here: the columns are orthonormal, meaning that they are orthogonal and each has 2-norm equal to 1. "
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Given any other vector of length 5, we can compute its inner product with each of the columns of $\\mathbf{Q}$. "
	]
	},
	{
	"cell_type": "code",
	"execution_count": 14,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"3x1 Array{Float64,2}:\n",
	" -1.11399 \n",
	" -0.901804 \n",
	" 0.00955111"
	]
	},
	"execution_count": 14,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"u = rand(5,1); c = Q'*u"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"We can then use these coefficients to find a vector in the column space of $\\mathbf{Q}$."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 15,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"5x1 Array{Float64,2}:\n",
	" 0.144826\n",
	" 0.796491\n",
	" 0.375148\n",
	" 0.8768 \n",
	" 0.699589"
	]
	},
	"execution_count": 15,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"v = Q*c"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"As explained in the text, $\\mathbf{r} = \\mathbf{u}-\\mathbf{v}$ is orthogonal to all of the columns of $\\mathbf{Q}$."
	]
	},
	{
	"cell_type": "code",
	"execution_count": 16,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"3x1 Array{Float64,2}:\n",
	" -2.24722e-16\n",
	" 7.41632e-17\n",
	" 1.13865e-16"
	]
	},
	"execution_count": 16,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"r = u-v; Q'*r"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"Consequently, we have decomposed $\\mathbf{u}=\\mathbf{v}+\\mathbf{r}$ into the sum of two orthogonal parts, one lying in the range of $\\mathbf{Q}$. "
	]
	},
	{
	"cell_type": "code",
	"execution_count": 17,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"1x1 Array{Float64,2}:\n",
	" 1.90887e-16"
	]
	},
	"execution_count": 17,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"v'*r"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"## Unitary matrices"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"We just saw that a matrix whose columns are orthonormal is pretty special. It becomes even more special if the matrix is also square, in which case we call it unitary. (In the real case, such matrices are confusingly called _orthogonal_. Ugh.) Say $\\mathbf{Q}$ is unitary and $m\\times m$. Then $\\mathbf{Q}^\\mathbf{Q}$ is an $m\\times m$ identity matrix---that is, $\\mathbf{Q}^=\\mathbf{Q}^{-1}$! It can't get much easier in terms of finding the inverse of a matrix. "
	]
	},
	{
	"cell_type": "code",
	"execution_count": 18,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"5x5 Array{Float64,2}:\n",
	" 2.77556e-16 5.55112e-17 1.57009e-16 3.10317e-16 1.75542e-16\n",
	" 1.24127e-16 2.48253e-16 3.04364e-16 1.41271e-16 1.24127e-16\n",
	" 3.88578e-16 2.77556e-16 3.60822e-16 1.57009e-16 1.49468e-16\n",
	" 1.77858e-16 1.19696e-16 2.22045e-16 1.24127e-16 8.32667e-17\n",
	" 2.48253e-16 1.75542e-16 1.57009e-16 2.65135e-16 1.19834e-16"
	]
	},
	"execution_count": 18,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"Q = full( qrfact(rand(5,5)+1im*rand(5,5))[:Q] );\n",
	"abs( inv(Q) - Q' )"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"The rank of $\\mathbf{Q}$ is $m$, so continuing the discussion above, the original vector $\\mathbf{u}$ lies in its column space. Hence the remainder $\\mathbf{r}=\\boldsymbol{0}$. "
	]
	},
	{
	"cell_type": "code",
	"execution_count": 19,
	"metadata": {
	"collapsed": false
	},
	"outputs": [
	{
	"data": {
	"text/plain": [
	"5x1 Array{Complex{Float64},2}:\n",
	" -1.94289e-16-7.63278e-17im\n",
	" 0.0+1.38778e-16im\n",
	" -2.77556e-16+2.35922e-16im\n",
	" -2.22045e-16-3.33067e-16im\n",
	" -2.22045e-16+1.11022e-16im"
	]
	},
	"execution_count": 19,
	"metadata": {},
	"output_type": "execute_result"
	}
	],
	"source": [
	"c = Q'*u; \n",
	"v = Q*c;\n",
	"r = u - v"
	]
	},
	{
	"cell_type": "markdown",
	"metadata": {},
	"source": [
	"This is another way to arrive at a fact we already knew: Multiplication by $\\mathbf{Q}^*=\\mathbf{Q}^{-1}$ changes the basis to the columns of $\\mathbf{Q}$."
	]
	}
	],
	"metadata": {
	"kernelspec": {
	"display_name": "Julia 0.4.3",
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	"file_extension": ".jl",
	"mimetype": "application/julia",
	"name": "julia",
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