Ch7_EigenValuesVector_Ex7.5.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "3bd754b6",
   "metadata": {},
   "source": [
    "<b>Question 1 </b>\n",
    "\n",
    "Determine the matrices $U, D$ and $V$ such that $A = UDV^T$ for the following :\n",
    "\n",
    "(a)$A = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&2\\\\\n",
    "\\end{pmatrix}\\quad$(b)$A = \\begin{pmatrix}\n",
    "1&4\\\\\n",
    "4&7\\\\\n",
    "\\end{pmatrix}\\quad$(c)$A = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "(d)$A = \\begin{pmatrix}\n",
    "1&0&1\\\\\n",
    "0&1&2\\\\\n",
    "\\end{pmatrix}\\quad$(e)$A = \\begin{pmatrix}\n",
    "1&1\\\\\n",
    "3&3\\\\\n",
    "\\end{pmatrix}\\quad$(f)$A = \\begin{pmatrix}\n",
    "1&1&1\\\\\n",
    "3&3&3\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "873e6a7d",
   "metadata": {},
   "source": [
    "<b> Solution (a) </b> \n",
    "\n",
    "Here, $A^TA = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&2\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow A^TA = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&4\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Eigenvalues of $A^TA: det(A - \\lambda I) = 0$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1- \\lambda&0\\\\\n",
    "0 & 4 - \\lambda\\\\\n",
    "\\end{pmatrix} = 0$\n",
    "\n",
    "$\\Rightarrow (1- \\lambda)(4 - \\lambda) = 0$\n",
    "\n",
    "$\\Rightarrow \\lambda_{1} =1$ and $\\lambda_{2} = 4$\n",
    "\n",
    "We know that, <br>\n",
    "If $A$ is $m$ by $n$ matrix and $\\lambda_{1}, \\lambda_{2},...,\\lambda_{n}$ be the eigenvalues of $A^TA$,, then singular values denoted by $\\sigma_{1}, \\sigma_{2},...\\sigma_{3}$ are the numbers such that: <br>\n",
    "$\\sigma_{1} = \\sqrt{\\lambda_{1}}, \\sigma_{2} = \\sqrt{\\lambda_{2}}$\n",
    "\n",
    "$\\Rightarrow$ <b>Here, singular values of $A$ are $\\sigma_{1} = \\sqrt{1} = 1$, and $\\sigma_{2} = \\sqrt{4} = 2$</b>\n",
    "\n",
    "$\\Rightarrow D = \\begin{pmatrix}\n",
    "\\sigma_{1}&0\\\\\n",
    "0&\\sigma_{2}\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next,<br>\n",
    "Eigenvector of $\\lambda_{1}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "1-1&0\\\\\n",
    "0&4-1\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "0&0\\\\\n",
    "0&3\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow y = 0$ and $x$ can be any non-zero $s$. Thus $v = \\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "and normalized eigenvector $v_{1} = \\frac{1}{\\lVert v \\rVert}v = \\frac{1}{1}v = \\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Eigenvector of $\\lambda_{2}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "1-4&0\\\\\n",
    "0&4-4\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-3&0\\\\\n",
    "0&0\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow x = 0$ and $y$ can be any non-zero $s$. Thus $v = \\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "and normalized eigenvector $v_{2} = \\frac{1}{\\lVert v \\rVert}v = \\frac{1}{1}v = \\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Therefore $V = (v_{1}\\quad v_{2})  =\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, <br>\n",
    "using $\\sigma_{1}u_{1} = Av_{1}$ and $\\sigma_{2}u_{2} = Av_{2}$\n",
    "\n",
    "we have, \n",
    "\n",
    "$u_{1} = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&2\\\\\n",
    "\\end{pmatrix}\\cdot\\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$2u_{1} = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&2\\\\\n",
    "\\end{pmatrix}\\cdot\\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{2} = \\frac{1}{2}\\begin{pmatrix}\n",
    "0\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "$\\Rightarrow U = (u_{1}\\quad u_{2}) = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Since, $U = V = I$<br>\n",
    "$A = UDV^T = IDI^T = A$\n",
    "\n",
    "\n",
    "<b> Solution (b) </b>\n",
    "\n",
    "Here, $A^TA = \\begin{pmatrix}\n",
    "1&4\\\\\n",
    "4&7\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "1&4\\\\\n",
    "4&7\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\qquad = \\begin{pmatrix}\n",
    "1+16&4+28\\\\\n",
    "4+28&16+49\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "17&32\\\\\n",
    "32&65\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "For Eigenvalues, $det(A - \\lambda I) = 0$\n",
    "we have, $\\begin{pmatrix}\n",
    "17-\\lambda&32\\\\\n",
    "32&65-\\lambda\\\\\n",
    "\\end{pmatrix} = 0$\n",
    "\n",
    "$\\Rightarrow (17 - \\lambda)(65 - \\lambda) -32(32) = 0$<br>\n",
    "\n",
    "$\\Rightarrow 1105 - 17\\lambda -65\\lambda +\\lambda^2 - 1024 = 0$<br>\n",
    "$\\Rightarrow \\lambda^2 - 82\\lambda + 81 = 0$<br>\n",
    "$\\Rightarrow \\lambda^2 - 81\\lambda -\\lambda + 81 = 0$<br>\n",
    "$\\Rightarrow \\lambda(\\lambda - 81) - 1(\\lambda - 81) =0$<br>\n",
    "$\\Rightarrow \\lambda = 81$ or $\\lambda = 1$\n",
    "\n",
    "\n",
    "$\\Rightarrow$ <b>Here, singular values of $A$ are $\\sigma_{1} = \\sqrt{81} = 9$, and $\\sigma_{2} = \\sqrt{1} = 1$</b>\n",
    "\n",
    "$\\Rightarrow D = \\begin{pmatrix}\n",
    "\\sigma_{1}&0\\\\\n",
    "0&\\sigma_{2}\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "9&0\\\\\n",
    "0&1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, eigenvector of $\\lambda_{1} = 81$\n",
    "with $(A - \\lambda I) u = 0$\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "17-81&32\\\\\n",
    "32&65-81\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-64&32\\\\\n",
    "32&-16\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -64x + 32y = 0$<br>\n",
    "and, $32x - 16y = 0$\n",
    "\n",
    "$\\Rightarrow -2x + y = 0$<br>\n",
    "and, $2x - y =0$\n",
    "\n",
    "$\\Rightarrow y = 2$ and $x = 1$ or eigenvector $v = \\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow$ normalized eigenvector, $v_{1} = \\frac{1}{\\lVert v \\rVert}v$<br>\n",
    "$\\Rightarrow v_{1} = \\frac{1}{\\lVert \\sqrt{1^2 + 2^2} \\rVert}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow v_{1} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Next, eigenvector of $\\lambda_{1} = 1$\n",
    "with $(A - \\lambda I) u = 0$\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "17-1&32\\\\\n",
    "32&65-1\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "16&32\\\\\n",
    "32&64\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow 16x + 32y = 0$<br>\n",
    "and, $32x + 64y = 0$\n",
    "\n",
    "$\\Rightarrow x + 2y = 0$<br>\n",
    "and, $x + 2y =0$\n",
    "\n",
    "$\\Rightarrow y = 1$ and $x = -2$ or eigenvector $v = \\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow$ normalized eigenvector, $v_{2} = \\frac{1}{\\lVert v \\rVert}v$<br>\n",
    "$\\Rightarrow v_{2} = \\frac{1}{\\lVert \\sqrt{(-2)^2 + 1^2} \\rVert}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow v_{2} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Therefore $V = (v_{1}\\quad v_{2})  =\\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1&-2\\\\\n",
    "2&1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, <br>\n",
    "using $\\sigma_{1}u_{1} = Av_{1}$ and $\\sigma_{2}u_{2} = Av_{2}$\n",
    "\n",
    "we have, <br>\n",
    "$9u_{1} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1&4\\\\\n",
    "4&7\\\\\n",
    "\\end{pmatrix}\\cdot \\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "9\\\\\n",
    "18\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{1} = \\frac{1}{9\\sqrt{5}}\\begin{pmatrix}\n",
    "9\\\\\n",
    "18\\\\\n",
    "\\end{pmatrix} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "and, $u_{2} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1&4\\\\\n",
    "4&7\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "2\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Therefore $U = (u_{1}\\quad u_{2})  =\\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1&2\\\\\n",
    "2&-1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "8703b86b",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Resultant matrix of UDV^T\n",
      "[[1. 4.]\n",
      " [4. 7.]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Define matrices U, D, and V\n",
    "U = np.array([[1, 2], [2, -1]]) / np.sqrt(5)\n",
    "D = np.array([[9, 0], [0, 1]])\n",
    "V = np.array([[1, -2], [2, 1]]) / np.sqrt(5)\n",
    "\n",
    "UDVT = np.matmul(np.matmul(U, D), V.T)\n",
    "\n",
    "# Round the elements to two decimal points\n",
    "result_rounded = np.round(UDVT, 2)\n",
    "\n",
    "print(\"Resultant matrix of UDV^T\")\n",
    "print(result_rounded)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3a619f0c",
   "metadata": {},
   "source": [
    "<b> Solution (c) </b> \n",
    "\n",
    "Here, $A^TA = \\begin{pmatrix}\n",
    "1&0&1\\\\\n",
    "0&1&2\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix}$<br>\n",
    "$\\qquad \\Rightarrow A^TA= \\begin{pmatrix}\n",
    "2&2\\\\\n",
    "2&5\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "For eigenvalues, $det(A - \\lambda I) = 0$<br>\n",
    "we have, <br>\n",
    "$\\begin{pmatrix}\n",
    "2-\\lambda&2\\\\\n",
    "2&5-\\lambda\\\\\n",
    "\\end{pmatrix} = 0$<br>\n",
    "\n",
    "$\\Rightarrow (2-\\lambda)(5-\\lambda) -2(2) = 0$<br>\n",
    "$\\Rightarrow 10 - 2\\lambda -5\\lambda + \\lambda^2 -4 = 0$<br>\n",
    "$\\Rightarrow \\lambda^2 -7\\lambda +6 = 0$<br>\n",
    "$\\Rightarrow \\lambda^2 -6\\lambda -\\lambda +6 = 0$<br>\n",
    "$\\Rightarrow \\lambda(\\lambda - 6) -1(\\lambda - 6) = 0$<br>\n",
    "$\\Rightarrow (\\lambda - 1)(\\lambda - 6) = 0$<br>\n",
    "\n",
    "$\\Rightarrow \\lambda_{1} = 1$ or $\\lambda_{2} = 6$ \n",
    "\n",
    "<b>Here, singular values of $A$ are $\\sigma_{1} = \\sqrt{1} = 1$ and $\\sigma_{2} = \\sqrt{6}$</b>\n",
    "\n",
    "$\\Rightarrow D = \\begin{pmatrix}\n",
    "\\sigma_{1}&0\\\\\n",
    "0&\\sigma_{2}\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&\\sqrt{6}\\\\\n",
    "0&0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Next, <br>\n",
    "eigenvector of $\\lambda_{1} = 1$ with $(A - \\lambda I)u = 0$<br>\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "2-1&2\\\\\n",
    "2&5-1\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1&2\\\\\n",
    "2&4\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow x + 2y = 0$<br>\n",
    "and, $2x + 4y = 0$\n",
    "\n",
    "$\\Rightarrow x= -2y$ or $x = -2$ and $y = 1$\n",
    "eigenvector, $v_{1} = \\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, normalized eigenvector <br>\n",
    "\n",
    "$v_{1} = \\frac{1}{\\lVert v_{1}\\rVert}v_{1} = \\frac{1}{\\lVert \\sqrt{(-2)^2 + 1^2}\\rVert}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow v_{1} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "Next, <br>\n",
    "eigenvector of $\\lambda_{2} = 6$ with $(A - \\lambda I)u = 0$<br>\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "2-6&2\\\\\n",
    "2&5-6\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "-4&2\\\\\n",
    "2&-1\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -4x + 2y = 0$<br>\n",
    "and, $2x - y = 0$\n",
    "\n",
    "$\\Rightarrow y= 2x$ or $x = 1$ and $y = 2$\n",
    "eigenvector, $v_{2} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, normalized eigenvector <br>\n",
    "\n",
    "$v_{2} = \\frac{1}{\\lVert v_{2}\\rVert}v_{2} = \\frac{1}{\\lVert \\sqrt{(1^2 + 2^2}\\rVert}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow v_{2} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Therefore, $V = (v_{1}\\qquad v_{2}) = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, using $\\sigma_{1}u_{1} = Av_{1}$ and $\\sigma_{2}u_{2} = Av_{2}$\n",
    "we have, <br>\n",
    "$1\\cdot u_{1} = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix}\\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\qquad = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$<br>\n",
    "$\\Rightarrow u_{1}= \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\sqrt{6}u_{2} =  \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix} \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{2} = \\frac{1}{\\sqrt{6}}\\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{2} = \\frac{1}{\\sqrt{30}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "We know that, <br>\n",
    "any given matrix $A$ of size $m$ by $n$ can be decomposed into $A = UDV^T$<br>\n",
    "where,<br>\n",
    "U -> $m \\times m$ orthogonal<br>\n",
    "D -> $m \\times n$ diagonal<br>\n",
    "V -> $n \\times n$ orthogonal<br>\n",
    "\n",
    "Thus, we need to find $u_{3}$ orthogonal to $u_{1}, u_{2}$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1&2&5\\\\\n",
    "-2&1&0\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow x+2y + 5z = 0$<br>\n",
    "and, $-2x + y = 0$\n",
    "\n",
    "$\\Rightarrow y = 2x$ and $x = -z$\n",
    "\n",
    "$\\Rightarrow x = 1, y =2$, and $z = -1$, and $u_{3} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "now, normalizing $u_{3}$\n",
    "\n",
    "$u_{3} = \\frac{1}{\\lVert u_{3}}u = \\frac{1}{\\sqrt{1^2 +2^2 + (-1)^2}} = \\frac{1}{\\sqrt{6}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Thus, we get orthogonal $U = (u_{1}\\quad u_{2}\\quad u_{3}) = \\begin{pmatrix}\n",
    "-2/\\sqrt{5}&1/\\sqrt{30}&1/\\sqrt{6}\\\\\n",
    "-1/\\sqrt{5}&2/\\sqrt{30}&2/\\sqrt{6}\\\\\n",
    "0&5/\\sqrt{30}&-1/\\sqrt{6}\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "8fc62193",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Resultant matrix of UDV^T\n",
      "[[ 1.  -0. ]\n",
      " [ 0.8  0.6]\n",
      " [ 1.   2. ]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "# Define the matrices\n",
    "U = np.array([\n",
    "    [-2/np.sqrt(5), 1/np.sqrt(30), 1/np.sqrt(6)],\n",
    "    [-1/np.sqrt(5), 2/np.sqrt(30), 2/np.sqrt(6)],\n",
    "    [0, 5/np.sqrt(30), -1/np.sqrt(6)]])\n",
    "\n",
    "D = np.array([[1, 0], [0, np.sqrt(6)], [0,0]])\n",
    "\n",
    "V = ((1/np.sqrt(5)) * np.array([[-2, 1],[1, 2]]))\n",
    "\n",
    "UDVT = np.matmul(np.matmul(U, D), V.T)\n",
    "\n",
    "# Round the elements to two decimal points\n",
    "result_rounded = np.round(UDVT, 1)\n",
    "\n",
    "print(\"Resultant matrix of UDV^T\")\n",
    "print(result_rounded)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4ff3fa61",
   "metadata": {},
   "source": [
    "<b> Solution (d)</b>\n",
    "\n",
    "Here, $A^TA = \\begin{pmatrix}\n",
    "1&0\\\\\n",
    "0&1\\\\\n",
    "1&2\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "1&0&1\\\\\n",
    "0&1&2\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "1&0&1\\\\\n",
    "0&1&2\\\\\n",
    "1&2&5\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "For eigenvalue, $det(A - \\lambda) = 0$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1-\\lambda&0&1\\\\\n",
    "0&1-\\lambda&2\\\\\n",
    "1&2&5-\\lambda\\\\\n",
    "\\end{pmatrix} = 0$\n",
    "\n",
    "$\\Rightarrow (1- \\lambda)[(1 - \\lambda)(5 - \\lambda) -4] - 0[0(5 - \\lambda) -1(2)] + 1[0(2) -1(1-\\lambda)] = 0$<br>\n",
    "\n",
    "$\\Rightarrow (1 - \\lambda)(5 - 5\\lambda - \\lambda + \\lambda^2 -4) -1 + \\lambda = 0$<br>\n",
    "\n",
    "$\\Rightarrow (1 - \\lambda)(\\lambda^2 - 6\\lambda + 1) -1 + \\lambda = 0$<br>\n",
    "\n",
    "$\\Rightarrow \\lambda^2 - 6\\lambda + 1 - \\lambda^3 + 6 \\lambda^2 -\\lambda -1 + \\lambda = 0$<br>\n",
    "\n",
    "$\\Rightarrow - \\lambda^3 + 7 \\lambda^2 -6 \\lambda = 0$<br>\n",
    "\n",
    "$\\Rightarrow \\lambda(\\lambda^2 - 7\\lambda + 6) = 0$<br>\n",
    "$\\Rightarrow \\lambda(\\lambda^2 -6\\lambda -\\lambda + 6) = 0$<br>\n",
    "$\\Rightarrow \\lambda(\\lambda -1)(\\lambda - 6) = 0$\n",
    "\n",
    "$\\Rightarrow \\lambda_{1} = 6, \\lambda_{2} = 1$ and $\\lambda_{3} = 0$\n",
    "\n",
    "We know that, <br>\n",
    "any given matrix $A$ of size $m$ by $n$ can be decomposed into $A = UDV^T$<br>\n",
    "where,<br>\n",
    "U -> $m \\times m$ orthogonal<br>\n",
    "D -> $m \\times n$ diagonal<br>\n",
    "V -> $n \\times n$ orthogonal<br>\n",
    "\n",
    "Therefore, singular values of $A$ has to be just two i.e, $\\sigma_{1} = \\sqrt{\\lambda_{1}} = \\sqrt{6}$ ,and $\\sigma_{2} = \\sqrt{\\lambda_{2}} = \\sqrt{1} = 1$\n",
    "\n",
    "Thus, diagonal matrix, $D = \\begin{pmatrix}\n",
    "\\sqrt{6}&0&0\\\\\n",
    "0&1&0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Next, <br>\n",
    "eigenvector for $\\lambda_{1} = 6$ with $(A - \\lambda I)u = 0$<br>\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "1-6&0&1\\\\\n",
    "0&1-6&2\\\\\n",
    "1&2&5-6\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "-5&0&1\\\\\n",
    "0&-5&2\\\\\n",
    "1&2&-1\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -5x + z = 0$<br>\n",
    "and, $-5y + 2z = 0$<br>\n",
    "and, $x + 2y -z = 0$\n",
    "\n",
    "$\\Rightarrow z = 5x$, and $y = 2x$<br>\n",
    "$\\Rightarrow x=1, y = 2$ and $z = 5$<br> So, eigenvector $v_{1} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, normalizing eigenvector, $v_{1} = \\frac{1}{\\lVert v_{1} \\rVert}v_{1} = \\frac{1}{\\lVert \\sqrt{1^2 + 2^2 + 5^2} \\rVert}v_{1} = \\frac{1}{\\sqrt{30}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, <br>\n",
    "eigenvector for $\\lambda_{2} = 1$ with $(A - \\lambda I)u = 0$<br>\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "1-1&0&1\\\\\n",
    "0&1-1&2\\\\\n",
    "1&2&5-1\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "0&0&1\\\\\n",
    "0&0&2\\\\\n",
    "1&2&4\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow z = 0$<br>\n",
    "and, $2z = 0$<br>\n",
    "and, $x + 2y + 4z = 0$\n",
    "\n",
    "$\\Rightarrow x = -2, y = 1$, and $z = 0$ and eigenvector $v_{2} = \\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, normalizing eigenvector, $v_{2} = \\frac{1}{\\lVert v_{2} \\rVert}v_{2} = \\frac{1}{\\lVert \\sqrt{(-2)^2 + 1^2 + 0^2} \\rVert}v_{2} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Next, <br>\n",
    "eigenvector for $\\lambda_{3} = 0$ with $(A - \\lambda I)u = 0$<br>\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "1-0&0&1\\\\\n",
    "0&1-0&2\\\\\n",
    "1&2&5\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "1&0&1\\\\\n",
    "0&1&2\\\\\n",
    "1&2&5\\\\\n",
    "\\end{pmatrix} \\cdot \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow x+ z = 0$<br>\n",
    "and, $y + 2z = 0$<br>\n",
    "and, $x + 2y + 5z = 0$\n",
    "\n",
    "$\\Rightarrow z = -x$ and $y = 2x$ So, $x =1, y =2$ and $z = -1$ and eigenvector $v_{3} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, normalizing eigenvector $v_{3} = \\frac{1}{\\lVert v_{3} \\rVert}v_{3} = \\frac{1}{\\lVert \\sqrt{1^2 + 2^2 + (-1)^2} \\rVert}v_{3} = \\frac{1}{\\sqrt{6}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "Therefore, $V = (v_{1}\\quad v_{2}\\quad v_{3}) = \\begin{pmatrix}\n",
    "1/\\sqrt{30}&-2/\\sqrt{5}&1/\\sqrt{6}\\\\\n",
    "2/\\sqrt{30}&1/\\sqrt{5}&2/\\sqrt{6}\\\\\n",
    "5/\\sqrt{30}&0&-1/\\sqrt{6}\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, using $\\sigma_{1}u_{1} = Av_{1}$ and $\\sigma_{2}u_{2} = Av_{2}$\n",
    "\n",
    "$\\Rightarrow \\sigma_{1}u_{1} = \\frac{1}{\\sqrt{30}}\\begin{pmatrix}\n",
    "1&0&1\\\\\n",
    "0&1&2\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "1\\\\\n",
    "2\\\\\n",
    "5\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "$u_{1} = \\frac{1}{\\sqrt{6}}\\frac{1}{\\sqrt{30}}\\begin{pmatrix}\n",
    "6\\\\\n",
    "12\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$u_{1} = \\frac{1}{\\sqrt{180}}\\begin{pmatrix}\n",
    "6\\\\\n",
    "12\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "next, <br>\n",
    "$\\sigma_{2}u_{2} = Av_{2}$<br>\n",
    "$\\Rightarrow u_{2} = \\begin{pmatrix}\n",
    "1&0&1\\\\\n",
    "0&1&2\\\\\n",
    "\\end{pmatrix}\\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{2} = \\frac{1}{\\sqrt{5}}\\begin{pmatrix}\n",
    "-2\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Therefore, $U = (u_{1} \\quad u_{2}) = \\begin{pmatrix}\n",
    "6/\\sqrt{180}&-2/\\sqrt{5}\\\\\n",
    "12/\\sqrt{180}&1/\\sqrt{5}\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "8d5f8597",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Resultant matrix of UDV^T\n",
      "[[ 1. -0.  1.]\n",
      " [-0.  1.  2.]]\n"
     ]
    }
   ],
   "source": [
    "U = np.array([[6/np.sqrt(180), -2/np.sqrt(5)],\n",
    "              [12/np.sqrt(180), 1/np.sqrt(5)]])\n",
    "\n",
    "D = np.array([[np.sqrt(6), 0, 0],\n",
    "              [0, 1, 0]])\n",
    "\n",
    "V = np.array([[1/np.sqrt(30), -2/np.sqrt(5), 1/np.sqrt(6)],\n",
    "              [2/np.sqrt(30), 1/np.sqrt(5), 2/np.sqrt(6)],\n",
    "              [5/np.sqrt(30), 0, -1/np.sqrt(6)]])\n",
    "\n",
    "UDVT = np.matmul(np.matmul(U, D), V.T)\n",
    "\n",
    "# Round the elements to two decimal points\n",
    "result_rounded = np.round(UDVT, 2)\n",
    "\n",
    "print(\"Resultant matrix of UDV^T\")\n",
    "print(result_rounded)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b18c5d73",
   "metadata": {},
   "source": [
    "<b> Solution (e) </b> \n",
    "\n",
    "Here, $A^TA = \\begin{pmatrix}\n",
    "1&3\\\\\n",
    "1&3\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "1&1\\\\\n",
    "3&3\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow A^TA = \\begin{pmatrix}\n",
    "10&10\\\\\n",
    "10&10\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Eigenvalues $det(A - \\lambda) = 0$<br>\n",
    "\n",
    "$\\Rightarrow (10 - \\lambda)(10 - \\lambda) -100 - 0$<br>\n",
    "$\\Rightarrow 100 - 10\\lambda -10\\lambda + \\lambda^2 - 100 = 0$<br>\n",
    "$\\Rightarrow \\lambda^2 - 20\\lambda = 0$<br>\n",
    "$\\Rightarrow \\lambda(\\lambda - 20) = 0$<br>\n",
    "\n",
    "$\\Rightarrow \\lambda_{1} = 20$ and $\\lambda_{2} = 0$\n",
    "\n",
    "Here, singular value of matrix $A$ are $\\sigma_{1} = \\sqrt{20}$ and $\\sigma_{2} = 0$\n",
    "\n",
    "Hence, diagonal matrix, $D = \\begin{pmatrix}\n",
    "\\sqrt{20}&0\\\\\n",
    "0&0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, eigenvector when $\\lambda_{1} = 20$ with $(A - \\lambda I)u = 0$<br>\n",
    "\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "10-20&10\\\\\n",
    "10&10-20\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow -10x + 10y = 0$<br>\n",
    "and, $10x -10y = 0$\n",
    "\n",
    "$\\Rightarrow x = y = 1$ and eigenvector $v_{1} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "now, normalizing eigenvector $v_{1} = \\frac{1}{\\lVert v_{1}\\rVert}v_{1} = \\frac{1}{\\lVert \\sqrt{1^2 + 1^2}\\rVert}v_{1} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "\n",
    "Next, eigenvector when $\\lambda_{2} = 0$ with $(A - \\lambda I)u = 0$<br>\n",
    "\n",
    "we get, <br>\n",
    "$\\begin{pmatrix}\n",
    "10-0&10\\\\\n",
    "10&10-0\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow 10x + 10y = 0$<br>\n",
    "and, $10x +10y = 0$\n",
    "\n",
    "$\\Rightarrow x = 1, y = -1$ and eigenvector $v_{2} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "now, normalizing eigenvector $v_{2} = \\frac{1}{\\lVert v_{2}\\rVert}v_{2} = \\frac{1}{\\lVert \\sqrt{1^2 + (-1)^2}\\rVert}v_{2} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Therefore, $V = (v_{1}\\quad v_{2}) = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1&1\\\\\n",
    "1&-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now using $\\sigma_{1}u_{1} = Av_{1}$\n",
    "\n",
    "$\\sqrt{20}u_{1} = \\begin{pmatrix}\n",
    "1&1\\\\\n",
    "3&3\\\\\n",
    "\\end{pmatrix}\\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{1} = \\frac{1}{\\sqrt{20}}\\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "1&1\\\\\n",
    "3&3\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{1} = \\frac{1}{\\sqrt{40}}\\begin{pmatrix}\n",
    "2\\\\\n",
    "6\\\\\n",
    "\\end{pmatrix} = \\frac{1}{\\sqrt{10}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "3\\\\\n",
    "\\end{pmatrix} $\n",
    "\n",
    "Next, to get, $u_{2}$ using, $u_{1}\\cdot u{2}  = 0$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "1/\\sqrt{10}\\\\\n",
    "3/\\sqrt{10}\\\\\n",
    "\\end{pmatrix}\\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow x = 3, y = -1$ and normalized $u_{2} = \\frac{1}{\\lVert \\sqrt{3^2 + (-1)^2}\\rVert}u_{2} = \\frac{1}{\\sqrt{10}}\\begin{pmatrix}\n",
    "3\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Therefore, $U = \\frac{1}{\\sqrt{10}}\\begin{pmatrix}\n",
    "1&3\\\\\n",
    "3&-1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "06c18f1f",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Resultant matrix of UDV^T\n",
      "[[1. 1.]\n",
      " [3. 3.]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "U = (1 / np.sqrt(10)) * np.array([[1, 3], [3, -1]])\n",
    "D = np.array([[np.sqrt(20), 0], [0, 0]])\n",
    "V = (1 / np.sqrt(2)) * np.array([[1, 1], [1, -1]])\n",
    "\n",
    "UDVT = np.matmul(np.matmul(U, D), V.T)\n",
    "\n",
    "# Round the elements to two decimal points\n",
    "result_rounded = np.round(UDVT, 2)\n",
    "\n",
    "print(\"Resultant matrix of UDV^T\")\n",
    "print(result_rounded)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0481a738",
   "metadata": {},
   "source": [
    "<b> Solution (f) </b>\n",
    "\n",
    "Here, $A^TA = \\begin{pmatrix}\n",
    "1&3\\\\\n",
    "1&3\\\\\n",
    "1&3\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "1&1&1\\\\\n",
    "3&3&3\\\\\n",
    "\\end{pmatrix}$ \n",
    "\n",
    "$\\Rightarrow A^TA = \\begin{pmatrix}\n",
    "10&10&10\\\\\n",
    "10&10&10\\\\\n",
    "10&10&10\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Eigenvalue of $A^TA: det(A - \\lambda I)  =0$\n",
    "\n",
    "$\\Rightarrow \\begin{pmatrix}\n",
    "10 - \\lambda&10&10\\\\\n",
    "10&10-\\lambda&10\\\\\n",
    "10&10&10-\\lambda\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow (10 - \\lambda)[(10 - \\lambda)(10 - \\lambda) -100] - 10[10(10 - \\lambda) -100] + 10[100 - 10(10 - \\lambda)] = 0$<br>\n",
    "$\\Rightarrow (10 - \\lambda)[100 - 10\\lambda - 10\\lambda + \\lambda^2 -100] - 10[100 - 10\\lambda -100] + 10[100 - 100 + 10 \\lambda)] = 0$<br>\n",
    "\n",
    "$\\Rightarrow (10 - \\lambda)(\\lambda^2 - 20\\lambda) + 100\\lambda + 100\\lambda = 0$<br>\n",
    "\n",
    "$\\Rightarrow 10\\lambda^2 -20\\lambda - \\lambda^3 + 20\\lambda^2 + 100\\lambda + 100\\lambda = 0$<br>\n",
    "\n",
    "$\\Rightarrow -\\lambda^3 + 30\\lambda^2 = 0$<br>\n",
    "$\\Rightarrow \\lambda^2(\\lambda - 30) = 0$\n",
    "\n",
    "\n",
    "$\\Rightarrow \\lambda_{1} = 30$ and $\\lambda_{2}= 0$, $\\lambda_{3} = 0$\n",
    "\n",
    "\n",
    "We know that, <br>\n",
    "any given matrix $A$ of size $m$ by $n$ can be decomposed into $A = UDV^T$<br>\n",
    "where,<br>\n",
    "U -> $m \\times m$ orthogonal<br>\n",
    "D -> $m \\times n$ diagonal<br>\n",
    "V -> $n \\times n$ orthogonal<br>\n",
    "\n",
    "<b>Here, $D$ would be of size $2 \\times 3$, where positive singular value of $A$, $\\sigma_{1} = \\sqrt{30}$</b>\n",
    "\n",
    "$\\Rightarrow D = \\begin{pmatrix}\n",
    "\\sqrt{30}&0&0\\\\\n",
    "0&0&0\\\\\n",
    "\\end{pmatrix}$\\\n",
    "\n",
    "\n",
    "Next, for eigenvector when $\\lambda_{1} = 30$ with $(A - \\lambda I)u = 0$ <br>\n",
    "we get, <br>\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "10-30&10&10\\\\\n",
    "10&10-30&10\\\\\n",
    "10&10&10-30\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "$\\Rightarrow -20x + 10y + 10z = 0$<br>\n",
    "and, $10x - 20y + 10z = 0$<br>\n",
    "and, $10x + 10y -20z = 0$<br>\n",
    "\n",
    "$\\Rightarrow x = y= z =1$<br>\n",
    "\n",
    "$v_{1} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, normalized eigenvector $v_{1} = \\frac{1}{\\lVert v_{1} \\rVert}v_{1} = \\frac{1}{\\sqrt{1^2 + 1^2 + 1^2}} = \\frac{1}{\\sqrt{3}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "Next, for eigenvector when $\\lambda_{2} = 0$ and $\\lambda_{3} = 0$ with $(A - \\lambda I)u = 0$ <br>\n",
    "we get, <br>\n",
    "\n",
    "$\\begin{pmatrix}\n",
    "10-0&10&10\\\\\n",
    "10&10-0&10\\\\\n",
    "10&10&10-0\\\\\n",
    "\\end{pmatrix} \\begin{pmatrix}\n",
    "x\\\\\n",
    "y\\\\\n",
    "z\\\\\n",
    "\\end{pmatrix} = \\begin{pmatrix}\n",
    "0\\\\\n",
    "0\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "\n",
    "$\\Rightarrow 10x + 10y + 10z = 0$<br>\n",
    "and, $10x +10y + 10z = 0$<br>\n",
    "and, $10x + 10y + 10z = 0$<br>\n",
    "\n",
    "This system of equation is degenrate, $x, y, z$ can be any set of numbers. However, as $U$ is needed to be orthogonal we pick values of $x, y, z$ for $v_{2}$ and $v_{3}$ such that $v_{2}\\cdot v_{3} = 0$\n",
    "\n",
    "If $v_{1} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "-1\\\\\n",
    "0\\\\\n",
    "\\end{pmatrix}$ we have $v_{2} = \\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "-2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Now, normalized eigenvector $v_{2} = \\frac{1}{\\lVert v_{2} \\rVert}v_{2} = \\frac{1}{\\sqrt{1^2 + (-1)^2 + 0^2}} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "-1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "and, normalized eigenvector $v_{3} = \\frac{1}{\\lVert v_{3} \\rVert}v_{3} = \\frac{1}{\\sqrt{1^2 + 1^2 + (-2)^2}} = \\frac{1}{\\sqrt{6}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "-2\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "<b>Therefore, $V = (v_{1}\\quad v_{2}\\quad v_{3}) = \\begin{pmatrix}\n",
    "1/\\sqrt{3}&1/\\sqrt{2}&1/\\sqrt{6}\\\\\n",
    "1/\\sqrt{3}&-1/\\sqrt{2}&1/\\sqrt{6}\\\\\n",
    "1/\\sqrt{3}&1/\\sqrt{2}&-2/\\sqrt{6}\\\\\n",
    "\\end{pmatrix}$</b>\n",
    "\n",
    "Now, using $\\sigma_{1}u_{1} = Av_{1}$\n",
    "\n",
    "$\\Rightarrow \\sqrt{30}u_{1} =\\begin{pmatrix}\n",
    "1&1&1\\\\\n",
    "3&3&3\\\\\n",
    "\\end{pmatrix} \\frac{1}{\\sqrt{3}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "1\\\\\n",
    "1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "$\\Rightarrow u_{1} = \\frac{1}{\\sqrt{30}}\\frac{1}{\\sqrt{3}}\\begin{pmatrix}\n",
    "3\\\\\n",
    "9\\\\\n",
    "\\end{pmatrix}$<br>\n",
    "\n",
    "\n",
    "$\\Rightarrow u_{1} = \\frac{1}{\\sqrt{90}}\\begin{pmatrix}\n",
    "3\\\\\n",
    "9\\\\\n",
    "\\end{pmatrix} = \\frac{1}{\\sqrt{10}}\\begin{pmatrix}\n",
    "1\\\\\n",
    "3\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "Next, $u_{2}$ needs to be orthogonal as well, Therefore $u_{2} = \\begin{pmatrix}\n",
    "3\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$ and normalized $u_{2} = \\frac{1}{\\sqrt{10}}\\begin{pmatrix}\n",
    "3\\\\\n",
    "-1\\\\\n",
    "\\end{pmatrix}$\n",
    "\n",
    "<b> Therefore, $U = (u_{1}\\quad u_{2}\\quad u_{3}) = \\frac{1}{\\sqrt{10}}\\begin{pmatrix}\n",
    "1&3\\\\\n",
    "3&-1\\\\\n",
    "\\end{pmatrix}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "6d3635f6",
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Resultant matrix of UDV^T\n",
      "[[1. 1. 1.]\n",
      " [3. 3. 3.]]\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "\n",
    "U = (1 / np.sqrt(10)) * np.array([[1, 3],\n",
    "                                  [3, -1]])\n",
    "\n",
    "D = np.array([[np.sqrt(30), 0, 0],\n",
    "              [0, 0, 0]])\n",
    "\n",
    "V = np.array([[1/np.sqrt(3), 1/np.sqrt(2), 1/np.sqrt(6)],\n",
    "              [1/np.sqrt(3), -1/np.sqrt(2), 1/np.sqrt(6)],\n",
    "              [1/np.sqrt(3), 1/np.sqrt(2), -2/np.sqrt(6)]])\n",
    "\n",
    "UDVT = np.matmul(np.matmul(U, D), V.T)\n",
    "\n",
    "# Round the elements to two decimal points\n",
    "result_rounded = np.round(UDVT, 2)\n",
    "\n",
    "print(\"Resultant matrix of UDV^T\")\n",
    "print(result_rounded)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "599d9cf2",
   "metadata": {},
   "source": [
    "------------------------------  "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fc5142db",
   "metadata": {},
   "source": [
    "<b> Question 2</b> \n",
    "\n",
    "Prove proposition 7.31"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "006f9aea",
   "metadata": {},
   "source": [
    "<b> Proposition 7.31: Let $A$ be any matrix. Then the eigenvalues of $A^TA$ are positive or zero</b>"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c25ea607",
   "metadata": {},
   "source": [
    "<b> Solution 2 </b> \n",
    "\n",
    "Let $\\lambda_{1}, \\lambda_{2}, \\cdots, \\lambda_{n}$ be the eigenvalues with eigenvectors $v_{1}, v_{2}, \\cdots, v_{n}$.\n",
    "\n",
    "Then, as we know $Av = \\lambda v$, so for any randomly chosen  $v_{j}$\n",
    "\n",
    "$A^TAv_{j} = \\lambda_{j}v_{j}$\n",
    "\n",
    "We also know that, <br>\n",
    "if $A$ is any matrix then $A^TA$ is a symmetric matrix.\n",
    "\n",
    "$\\Rightarrow A^TA$ is orthogonally diagonalizable<br>\n",
    "$\\Rightarrow$ eigenvectors are orthonormal\n",
    "\n",
    "Now taking norm square:<br>\n",
    "\n",
    "$\\lVert Av_{j} \\rVert^2 = Av_{j}\\cdot Av_{j}$\n",
    "\n",
    "$\\qquad = (Av_{j})^T Av_{j}\\qquad$ (Since $(u\\cdot v = u^T\\cdot v))$\n",
    "$\\qquad = A^T V_{j}^T Av_{j}$<br>\n",
    "$\\qquad = v_{j}^T A^TAv_{j}$<br>\n",
    "$\\qquad = v_{j}^T \\lambda_{j}v_{j}$<br>\n",
    "$\\qquad = \\lambda_{j} (v_{j}\\cdot v_{j}) = \\lambda_{j}$<br>\n",
    "\n",
    "Now, we have $\\lambda_{j} = \\lVert Av_{j} \\rVert^2$ i.e., $\\ge 0$\n",
    "\n",
    "$\\Rightarrow \\lambda_{j} \\ge 0$<br>\n",
    "\n",
    "Hence, all eigenvalues of $A^TA$ are either positive values or zeroes."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35126538",
   "metadata": {},
   "source": [
    "-------------------------------   "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "76d40823",
   "metadata": {},
   "source": [
    "<b> Question 3</b> \n",
    "\n",
    "Let matrix $A$ have $k-$positive singular values. Show that rank of matrix $A$ is $k$\n",
    "\n",
    "Hint: <br>\n",
    "If matrix $X$ is invertible then $rank(XA) = rank(A)$ and $rank(AX) = rank(A)$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2a626890",
   "metadata": {},
   "source": [
    "<b> Solution 3</b>\n",
    "\n",
    "We know that, <br>\n",
    "any given matrix $A$ of size $m$ by $n$ by SVD can be decomposed into $A = UDV^T$<br>\n",
    "where,<br>\n",
    "U -> $m \\times m$ orthogonal<br>\n",
    "D -> $m \\times n$ diagonal<br>\n",
    "V -> $n \\times n$ orthogonal<br>\n",
    "\n",
    "Now, since $U$ is orthogonal, it is invertible<br>\n",
    "and, $U^{-1} = U^T$\n",
    "\n",
    "Similarly, $V^{-1} = V^T$<br>\n",
    "$\\Rightarrow (V^T)^{-1} = V$<br>\n",
    "$\\Rightarrow V^T$ is invertible\n",
    "\n",
    "We are given, <br>\n",
    "If matrix $X$ is invertible then $rank(XA) = rank(A)$ and $rank(AX) = rank(A)$\n",
    "\n",
    "$\\Rightarrow rank(A) = rank(UDV^T)$<br>\n",
    "$\\qquad = rank(DV^T)$<br>\n",
    "$\\qquad = rank(D)$<br>\n",
    "$\\qquad = k$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "96059af4",
   "metadata": {},
   "source": [
    "-----------------------------    "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "fb2fc016",
   "metadata": {},
   "source": [
    "<b> Question 4</b> \n",
    "\n",
    "Prove proposition 7.33 "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f95e6fe6",
   "metadata": {},
   "source": [
    "<b> Proposition 7.33: Let $A$ be an $m$ by $n$ matrix with factorization given by $A = UDV^T$. Let matrix $A$ have $k \\le n$ positive singular values. Then we have the following:<br>\n",
    "   (a) The set of vectors $\\{u_{1}, u_{2}, \\cdots, u_{k}\\}$ form an orthonormal basis for the column space of matrix $A$.<br>\n",
    "    (b) The set of vectors $\\{v_{1}, v_{2}, \\cdots, v_{k}\\}$ form an orthonormal basis for the row space of matrix $A$<br>\n",
    "    (c) The set of vectors $\\{v_{k+1}, v_{K+2}, \\cdots, v_{n}\\}$ form an orthonormal basis for the null space of matrix $A$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4dd35753",
   "metadata": {},
   "source": [
    "---------------------------- "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf889dff",
   "metadata": {},
   "source": [
    "<b> Question 5 </b> \n",
    "\n",
    "Prove that the singular values $\\sigma_{1}, \\sigma_{2}, \\dots \\sigma_{n}$ of an $m$ by $n$ matrix $A$ are unique. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "53dcc585",
   "metadata": {},
   "source": [
    "<b> Solution 5</b> \n",
    "\n",
    "To get singular values of an $m$ by $n$ matrix $A$, we take eigenvalues of $A^TA$ and thereby, singular values as square roots.\n",
    "\n",
    "As, $A^TA$ makes it a $n$ by $n$ matrix, and we know that, for a square matrix, $A$ with $\\lambda$ eigenvalues and $u$ eigenvector. The eigenvalue $\\lambda$ is unique.\n",
    "\n",
    "Therefore, singular values or square roots of those eigenvalues must also be unique. \n",
    "\n",
    "Hence, proved"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e45dc550",
   "metadata": {},
   "source": [
    "---------------------------------- "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "741f7a89",
   "metadata": {},
   "source": [
    "<b> Question 6</b> \n",
    "\n",
    "Prove that the column vectors of the orthogonal matrix $U$ in $A = UDV^T$ are the eigenvectors of $AA^T$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b0d47481",
   "metadata": {},
   "source": [
    "<b> Solution </b>\n",
    "\n",
    "By $SVD$, we have<br>\n",
    "\n",
    "$A  = UDV^T$\n",
    "\n",
    "$\\Rightarrow AA^T = UDV^T (UDV^T)^T$<br>\n",
    "\n",
    "$\\qquad = (UDV^T) (V^T)^T U^TD^T$\n",
    "\n",
    "$\\qquad = UD\\cdot V^TV \\cdot U^TD^T$\n",
    "\n",
    "$\\qquad = U (DD^T)U^T$\n",
    "\n",
    "if $D^{'} = DD^T$<br>\n",
    "then, $AA^T = UD^{'}U^T$\n",
    "\n",
    "Since, $U$ is orthogonal, its inverse is $U^T$\n",
    "\n",
    "Next, <br>\n",
    "\n",
    "$U^T(AA^T)U = U^TU \\cdot (D^{'}) \\cdot U^TU$\n",
    "\n",
    "$\\Rightarrow U^T (AA^T) U = D^{'}$\n",
    "\n",
    "$\\Rightarrow U$ diagonalizes $AA^T$ and the columns of $A$ are eigenvectors of $AA^T$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "93d1d0ea",
   "metadata": {},
   "source": [
    "------------------------- "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "060c85fb",
   "metadata": {},
   "source": [
    "<b> Question 7</b> \n",
    "\n",
    "Prve that singular values of $A$ and $A^T$ are identical"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "33512456",
   "metadata": {},
   "source": [
    "<b> Solution 7 </b>\n",
    "\n",
    "The singular values of a matrix $A$ are given by the square roots of the eigenvalues of $A^TA$: $\\lambda_{1}, \\lambda_{2},\\cdots, \\lambda_{n}$\n",
    "\n",
    "Similarly, singular values of a matrix $A^T$ are given by the sqaure roots of the eigenvalues of $(A^T)^TA^T$: $\\delta_{1}, \\delta_{2}, \\cdots, \\delta_{n}$\n",
    "\n",
    "We know that, <br>\n",
    "\n",
    "The eigenvalues of the transposed matrix, $A^T$, are exactly the eigenvlues of $A$.\n",
    "\n",
    "$\\Rightarrow$ eigenvalues of transposed matrix $(A^T)^TA^T = $eigenvalues of $A^TA$\n",
    "\n",
    "$\\Rightarrow \\delta_{1} = \\lambda_{1}, \\delta_{2} = \\lambda_{2}, \\cdots, \\delta_{n} = \\lambda_{n}$\n",
    "\n",
    "$\\Rightarrow \\sqrt{\\delta_{1}} = \\sqrt{\\lambda_{1}}, \\sqrt{\\delta_{2}} = \\sqrt{\\lambda_{2}}, \\cdots, \\sqrt{\\delta_{n}} = \\sqrt{\\lambda_{n}}$\n",
    "\n",
    "Hence, proved singular value of $A^T$ and $A$ are equal"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b70f922a",
   "metadata": {},
   "source": [
    "--------------------- "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a39571ec",
   "metadata": {},
   "source": [
    "<b> Question 8 </b> \n",
    "\n",
    "Let $T: \\mathbb{R}^n \\to \\mathbb{R}^m$ be a linear transformation given by $T(x) = Ax$, where $A$ is an $m$ by $n$ matrix. Let singular value decomposition $A = UDV^T$ with $k \\le n$ positive singular values of matrix $A$. Prove the following results: \n",
    "\n",
    "(a) The set of vectors $u_{1}, u_{2}, \\dots u_{k}$ form an orthogonal basis for the range of $T$.<br>\n",
    "(b) The set of vectors $v_{k+1}, v_{k+2},\\dots v_{n}$ form an orthogonal basis for the kernel of $T$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "05b1f149",
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.9.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}