{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "using LinearAlgebra"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Square Case first"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{Float64,2}:\n",
       " 0.330869  0.843845  0.0623888  0.398208\n",
       " 0.8661    0.204488  0.138221   0.218923\n",
       " 0.741009  0.775278  0.576722   0.9775  \n",
       " 0.87276   0.139498  0.072938   0.983904"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(4,4)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}\n",
       "Q factor:\n",
       "4×4 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:\n",
       " -0.224588   0.76522    0.603058  -0.017861 \n",
       " -0.587892  -0.292547   0.130273  -0.742852 \n",
       " -0.502982   0.437253  -0.739298   0.0962124\n",
       " -0.592413  -0.371031   0.269792   0.662266 \n",
       "R factor:\n",
       "4×4 Array{Float64,2}:\n",
       " -1.47323  -0.782326  -0.428561  -1.29268 \n",
       "  0.0       0.873139   0.232416   0.303028\n",
       "  0.0       0.0       -0.351061  -0.188552\n",
       "  0.0       0.0        0.0        0.575913"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q,R  = qr(A)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 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"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round.(Q'Q, digits=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 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"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round.(Q*Q', digits=0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## I'm happy, I see Q is orthogonal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A ≈ Q * R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{Float64,2}:\n",
       " 0.330869  0.843845  0.0623888  0.398208\n",
       " 0.8661    0.204488  0.138221   0.218923\n",
       " 0.741009  0.775278  0.576722   0.9775  \n",
       " 0.87276   0.139498  0.072938   0.983904"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q * R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{Float64,2}:\n",
       " 0.330869  0.843845  0.0623888  0.398208\n",
       " 0.8661    0.204488  0.138221   0.218923\n",
       " 0.741009  0.775278  0.576722   0.9775  \n",
       " 0.87276   0.139498  0.072938   0.983904"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now a tall skinny example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×3 Array{Float64,2}:\n",
       " 0.657045  0.214426  0.0462177\n",
       " 0.268263  0.208357  0.269215 \n",
       " 0.410459  0.948475  0.756601 \n",
       " 0.391947  0.683485  0.28925  \n",
       " 0.949405  0.999374  0.570501 "
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A = rand(5,3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×3 Array{Float64,2}:\n",
       " -0.499955  -0.62267     0.00434913\n",
       " -0.204125  -0.0946497  -0.628421  \n",
       " -0.312324   0.686918   -0.424575  \n",
       " -0.298239   0.362414    0.645285  \n",
       " -0.722416   0.0110753   0.0917176 "
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q,R = qr(A)\n",
    "Q = Q[:,1:3] # make sure we have the first three columns"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×3 Array{Float64,2}:\n",
       " -0.499955  -0.62267     0.00434913\n",
       " -0.204125  -0.0946497  -0.628421  \n",
       " -0.312324   0.686918   -0.424575  \n",
       " -0.298239   0.362414    0.645285  \n",
       " -0.722416   0.0110753   0.0917176 "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       " -1.31421  -1.37177  -0.81277 \n",
       "  0.0       0.75706   0.57661 \n",
       "  0.0       0.0      -0.251239"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×3 Array{Float64,2}:\n",
       " 0.657045  0.214426  0.0462177\n",
       " 0.268263  0.208357  0.269215 \n",
       " 0.410459  0.948475  0.756601 \n",
       " 0.391947  0.683485  0.28925  \n",
       " 0.949405  0.999374  0.570501 "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q * R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×3 Array{Float64,2}:\n",
       " 0.657045  0.214426  0.0462177\n",
       " 0.268263  0.208357  0.269215 \n",
       " 0.410459  0.948475  0.756601 \n",
       " 0.391947  0.683485  0.28925  \n",
       " 0.949405  0.999374  0.570501 "
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "true"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "A ≈ Q * R"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "3×3 Array{Float64,2}:\n",
       "  1.0  -0.0  -0.0\n",
       " -0.0   1.0  -0.0\n",
       " -0.0  -0.0   1.0"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "round.(Q'Q, digits=0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5×5 Array{Float64,2}:\n",
       "  0.637693    0.158256   -0.273422   -0.0737523  0.354678 \n",
       "  0.158256    0.445538    0.265548   -0.378935   0.0887777\n",
       " -0.273422    0.265548    0.749666    0.0681243  0.194295 \n",
       " -0.0737523  -0.378935    0.0681243   0.636683   0.27865  \n",
       "  0.354678    0.0887777   0.194295    0.27865    0.53042  "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Q * Q'"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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