TB Lecture 6: Projectors

TB-Lecture-06-julia.ipynb

TB-Lecture-06-matlab.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 6: Projectors"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Orthogonality in projection"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the map on $\\mathbb{R}^2$ defined by"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P =\n",
      "\n",
      "     0     0\n",
      "     1     1\n"
     ]
    }
   ],
   "source": [
    "P = [0 0; 1 1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The result of applying $P$ is to zero out the first component and replace the second component with the sum of the coordinates. For a vector on the $x_2$-axis, this operation leaves the vector unchanged, i.e.,"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "     0\n"
     ]
    }
   ],
   "source": [
    "norm(P^2-P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus $P$ is a projector. As shown in the text, it can't be an orthogonal projector because it's nonsymmetric. It may help to visualize the map."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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CiEACNxhLs48zKeUh74yrGG8jpfdD8l8mIY2ooVwYEUjguJQGfT7OpPR+SOmr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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "[X,Y] = meshgrid(-5:5);\n",
    "x = [X(:) Y(:)];  Px = (P*x')';\n",
    "plot([x(:,1) Px(:,1)]',[x(:,2) Px(:,2)]','b'), axis equal\n",
    "hold on\n",
    "plot(Px(:,1),Px(:,2),'ko')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The direction of \"motion\" in the mapping is not orthogonal to the range of the map. You can also see that the singular values are not just 1 and 0, as is required of an orthogonal projector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "   1.4142e+00\n",
      "            0\n"
     ]
    }
   ],
   "source": [
    "svd(P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The orthogonal projector onto the same space is very boring."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "P =\n",
      "\n",
      "     0     0\n",
      "     0     1\n"
     ]
    }
   ],
   "source": [
    "P = [0 0; 0 1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The map is equally boring. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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vJjoD4JFdioJA2uut0JuMqURR7I7K5VgpPa1PtWpcu1wusnVDb8vns65gQWGV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\nFwgkAIAXCCQAgBcIJACAFwgkAIAX/j+UfAT3w7fZEQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = [X(:) Y(:)];  Px = (P*x')';\n",
    "plot([x(:,1) Px(:,1)]',[x(:,2) Px(:,2)]','b'), axis equal\n",
    "hold on\n",
    "plot(Px(:,1),Px(:,2),'ko')"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Matlab",
   "language": "matlab",
   "name": "matlab"
  },
  "language_info": {
   "codemirror_mode": "octave",
   "file_extension": ".m",
   "help_links": [
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-octave",
   "name": "matlab",
   "version": "0.11.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 1
}