Trefethen & Bau lecture notes in MATLAB

TB-Lecture-01.ipynb

TB-Lecture-02.ipynb

TB-Lecture-03.ipynb

TB-Lecture-04.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 4: Singular value decomposition"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Geometric interpretation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The SVD has many possible interpretations and uses. We can visualize the geometry of the SVD using the same 2D picture used to illustrate the induced 2-norm."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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aKPqNcRPq5VCAd7eajmIBrfX8Vx47n1EbRbGSVq5ppqTVdI5Bsm8hPc/eAF0h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\niFjC/weki7dh+zjn+QAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "A = [-2 0; -1 3];\n",
    "t = linspace(0,2*pi,300);\n",
    "x1 = cos(t);  x2 = sin(t);\n",
    "x = [x1;x2];  Ax = A*x;\n",
    "subplot(121), scatter(x1,x2,4,t,'.'), axis equal, axis off, title('x')\n",
    "subplot(122), scatter(Ax(1,:),Ax(2,:),4,t,'.'), axis equal, axis off, title('Ax')\n",
    "colormap(hsv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **left singular vectors** and their associated **singular values** are found from the major and minor axes of the ellipse. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "sigma =\n",
      "\n",
      "    3.2566    1.8424\n",
      "\n",
      "\n",
      "U =\n",
      "\n",
      "    0.2883    0.9558\n",
      "    0.9576   -0.2939\n"
     ]
    }
   ],
   "source": [
    "normAx = sqrt( sum(Ax.^2,1) );  % 2-norms of all the vectors\n",
    "[sigma1,k1] = max(normAx);  [sigma2,k2] = min(normAx);\n",
    "sigma = [ sigma1 sigma2 ]\n",
    "U = [ Ax(:,k1)/sigma1 Ax(:,k2)/sigma2 ]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The **right singular vectors** are the pre-images of those vectors on the ellipse."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "V =\n",
      "\n",
      "   -0.4694   -0.8805\n",
      "    0.8830   -0.4740\n"
     ]
    }
   ],
   "source": [
    "V = [ x(:,k1) x(:,k2) ]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Xbh1XjUaj0DgydHgv1izEb1W2b/eRGrS3YHqW6ByCue00pqqGqgYgFZDy+15d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\nRU5ErkJggg==\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "subplot(121), scatter(x1,x2,4,t,'.'), axis equal, axis off, title('x')\n",
    "hold on, plot([0 0; V(1,:)],[0 0; V(2,:)], 'k' )\n",
    "text(1.2*V(1,:),1.2*V(2,:),{'v_1','v_2'})\n",
    "subplot(122), scatter(Ax(1,:),Ax(2,:),4,t,'.'), axis equal, axis off, title('Ax')\n",
    "hold on, plot([0 0; U(1,:).*sigma],[0 0; U(2,:).*sigma], 'k' )\n",
    "text(1.1*U(1,:).*sigma,1.1*U(2,:).*sigma,{'\\sigma_1 u_1','\\sigma_2 u_2'})\n",
    "colormap(hsv)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Having searched over only a finite number of vectors, we have approximately (up to choices of signs in the singular vectors) found the SVD of this matrix."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U =\n",
      "\n",
      "    0.2898    0.9571\n",
      "    0.9571   -0.2898\n",
      "\n",
      "\n",
      "S =\n",
      "\n",
      "    3.2566         0\n",
      "         0    1.8424\n",
      "\n",
      "\n",
      "V =\n",
      "\n",
      "   -0.4719   -0.8817\n",
      "    0.8817   -0.4719\n"
     ]
    }
   ],
   "source": [
    "[U,S,V] = svd(A)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Full versus thin"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the case of a rectangular matrix, some of the SVD is essentially dead weight. We consider (as always in this text) the case with $m>n$, a tall and skinny matrix. The range of $A$ can have at most only $n$ dimensions. The last $m-n$ columns of $U$ therefore describe the remainder of $\\mathbb{C}^m$ in which the range is embedded (i.e., the orthogonal complement). That description can be changed without affecting the matrix at all.\n",
    "\n",
    "Algebraically, we have \n",
    "\n",
    "$$U = \\bigl[ U_1 \\: U_2 \\bigr], \\qquad S = \\begin{bmatrix} S_1 \\\\ 0 \\end{bmatrix},$$\n",
    "\n",
    "where the breaks occur after $n$ columns in $U$ and $n$ rows in $S$. Therefore $US=U_1S_1$, and we change nothing by replacing the SVD with $U_1S_1V^*$. The book calls this the **reduced SVD**, though it is also called the **thin SVD**. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "U =\n",
      "\n",
      "  -4.2591e-01  -4.2726e-01  -1.9420e-01   7.7352e-01\n",
      "  -3.6192e-01   1.8280e-01   9.0498e-01   1.2889e-01\n",
      "  -6.8998e-01  -3.6876e-01  -1.1424e-01  -6.1229e-01\n",
      "  -4.5994e-01   8.0501e-01  -3.6090e-01   1.0080e-01\n",
      "\n",
      "\n",
      "S =\n",
      "\n",
      "   1.8437e+00            0            0\n",
      "            0   7.1187e-01            0\n",
      "            0            0   1.7062e-01\n",
      "            0            0            0\n",
      "\n",
      "\n",
      "V =\n",
      "\n",
      "  -5.8697e-01  -7.8890e-01   1.8195e-01\n",
      "  -6.6113e-01   3.3734e-01  -6.7016e-01\n",
      "  -4.6731e-01   5.1366e-01   7.1957e-01\n",
      "\n",
      "\n",
      "U1 =\n",
      "\n",
      "  -4.2591e-01  -4.2726e-01  -1.9420e-01\n",
      "  -3.6192e-01   1.8280e-01   9.0498e-01\n",
      "  -6.8998e-01  -3.6876e-01  -1.1424e-01\n",
      "  -4.5994e-01   8.0501e-01  -3.6090e-01\n",
      "\n",
      "\n",
      "S1 =\n",
      "\n",
      "   1.8437e+00            0            0\n",
      "            0   7.1187e-01            0\n",
      "            0            0   1.7062e-01\n",
      "\n",
      "\n",
      "V =\n",
      "\n",
      "  -5.8697e-01  -7.8890e-01   1.8195e-01\n",
      "  -6.6113e-01   3.3734e-01  -6.7016e-01\n",
      "  -4.6731e-01   5.1366e-01   7.1957e-01\n"
     ]
    }
   ],
   "source": [
    "A = rand(4,3); \n",
    "[U,S,V] = svd(A)       % full SVD\n",
    "[U1,S1,V] = svd(A,0)   % thin SVD"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The price we pay is that while the columns of $U_1$ are orthonormal, we can't call it a unitary matrix because it isn't square. Take note:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ans =\n",
      "\n",
      "    1.0000    0.0000    0.0000\n",
      "    0.0000    1.0000   -0.0000\n",
      "    0.0000   -0.0000    1.0000\n",
      "\n",
      "\n",
      "ans =\n",
      "\n",
      "    0.6637    0.2461    0.0184   -0.1936    0.3450    0.0761\n",
      "    0.2461    0.3066   -0.0103    0.3264    0.2112   -0.0286\n",
      "    0.0184   -0.0103    0.4509    0.0661   -0.0694    0.4878\n",
      "   -0.1936    0.3264    0.0661    0.8162    0.0367   -0.0168\n",
      "    0.3450    0.2112   -0.0694    0.0367    0.2232   -0.0599\n",
      "    0.0761   -0.0286    0.4878   -0.0168   -0.0599    0.5394\n"
     ]
    }
   ],
   "source": [
    "U1'*U1    % n by n identity\n",
    "U1*U1'    % NOT the m by m identity"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll discuss the significance of $U_1U_1^*$ in due course."
   ]
  }
 ],
 "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
}

TB-Lecture-05.ipynb

TB-Lecture-06.ipynb

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