final version.

.ipynb_checkpoints/Exercise 3.3-checkpoint.ipynb

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+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "# Exercise 3.3\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Vector and matrix p-norms are related by various inequalities, often involving the dimensions m or n. \n",
+    "\n",
+    "For each of the following, verify the inequality and give an example of a nonzero vector or matrix(for general m,n) for which equality is achieved.\n",
+    "\n",
+    "In this problem x is an m-vector and A is an m x n matrix."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## (a) $$ ||x||_\\infty \\leq ||x||_2 $$\n",
+    "\n",
+    "$$ \\Vert x \\Vert_\\infty $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Lets consider a vector x; \n",
+    "\\begin{equation}\n",
+    "    x\n",
+    "     =\n",
+    "    \\begin{bmatrix}\n",
+    "    1 \\\\\n",
+    "    2 \\\\\n",
+    "    3\n",
+    "    \\end{bmatrix}\n",
+    "\\end{equation}\n",
+    "\n",
+    "Here, \n",
+    "\\begin{equation*}\n",
+    "||x||_\\infty = max |x_i| , where (1\\leq i \\leq m) = 3 \n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "||x||_2 = (\\sum_{i=1}^m |x_i|^2)^{1/2} = 3.742\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "3 \\leq 3.742\n",
+    "\\end{equation*}\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### (b)  $$||x||_2 \\leq \\sqrt{m}||x||_\\infty $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "\\begin{equation*}\n",
+    "\\sqrt{m}||x||_\\infty = 1.732*3 = 5.196\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "3.742 \\leq 5.196\n",
+    "\\end{equation*}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### (c)  $$||A||_\\infty \\leq \\sqrt{n}||A||_2 $$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### (d) $$||A||_2 \\leq \\sqrt{m}||A||_\\infty $$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Octave",
+   "language": "octave",
+   "name": "octave"
+  },
+  "language_info": {
+   "file_extension": ".m",
+   "help_links": [
+    {
+     "text": "GNU Octave",
+     "url": "https://www.gnu.org/software/octave/support.html"
+    },
+    {
+     "text": "Octave Kernel",
+     "url": "https://github.com/Calysto/octave_kernel"
+    },
+    {
+     "text": "MetaKernel Magics",
+     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+    }
+   ],
+   "mimetype": "text/x-octave",
+   "name": "octave",
+   "version": "4.2.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}

Exercise 3.3.ipynb

@@ -0,0 +1,247 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "# Exercise 3.3\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "Vector and matrix p-norms are related by various inequalities, often involving the dimensions m or n. \n",
+    "\n",
+    "For each of the following, verify the inequality and give an example of a nonzero vector or matrix(for general m,n) for which equality is achieved.\n",
+    "\n",
+    "In this problem x is an m-vector and A is an m x n matrix."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "## (a) $$ \\Vert x\\Vert_\\infty \\leq \\Vert x\\Vert_2 $$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "$$ \\Vert x\\Vert_\\infty = max |x_i| , where 1 \\leq i \\leq m$$\n",
+    "$$ \\Vert x\\Vert_2 = (\\sum_{i=1}^m |x_i|^2)^{1/2} $$\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "max |x_i| = max (|x_i|^2)^{1/2} \\leq (\\sum_{i=1}^m |x_i|^2)^{1/2}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "So, it can be seen that;\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\Vert x\\Vert_\\infty \\leq \\Vert x\\Vert_2\n",
+    "\\end{equation*}\n",
+    "\n",
+    "For example, lets consider a vector x; \n",
+    "\\begin{equation*}\n",
+    "    x\n",
+    "     =\n",
+    "    \\begin{bmatrix}\n",
+    "    1 \\\\\n",
+    "    2 \\\\\n",
+    "    3\n",
+    "    \\end{bmatrix}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\Vert x\\Vert_\\infty = 3 \n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\Vert x\\Vert_2 = 3.742\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "3 \\leq 3.742\n",
+    "\\end{equation*}\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "## (b)  $$\\Vert x\\Vert_2 \\leq \\sqrt{m}\\Vert x\\Vert_\\infty $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "This can be verified by following; \n",
+    "\n",
+    "\\begin{equation*}\n",
+    "(\\sum_{i=1}^m |x_i|^2)^{1/2} \\leq (\\sum_{i=1}^m max (|x_i|^2))^{1/2} = (m\\Vert x\\Vert_\\infty^2)^{1/2} = \\sqrt{m}\\Vert x\\Vert_\\infty\n",
+    "\\end{equation*}\n",
+    "\n",
+    "For the same x vector above, \n",
+    "\n",
+    "\\begin{equation*}\n",
+    "\\sqrt{m}\\Vert x\\Vert_\\infty = 1.732*3 = 5.196\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "3.742 \\leq 5.196\n",
+    "\\end{equation*}"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "## (c)  $$\\Vert A\\Vert_\\infty \\leq \\sqrt{n}\\Vert A\\Vert_2 $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "\\begin{equation*}\n",
+    "\\Vert A\\Vert_\\infty = sup \\frac{\\Vert Ax\\Vert_\\infty}{\\Vert x\\Vert_\\infty}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "Since we know that $ \\Vert x\\Vert_\\infty \\leq \\Vert x\\Vert_2 $;\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "sup \\frac{\\Vert Ax\\Vert_\\infty}{\\Vert x\\Vert_\\infty} \\leq sup \\frac{\\Vert Ax\\Vert_2}{\\Vert x\\Vert_\\infty}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\n",
+    "As well, since $ \\Vert x\\Vert_2 \\leq \\sqrt{m}\\Vert x\\Vert_\\infty $ and $x$ is in $\\mathbb{C}^n$ ;\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "sup \\frac{\\Vert Ax\\Vert_2}{\\Vert x\\Vert_\\infty} \\leq sup \\frac{\\Vert Ax\\Vert_2}{\\Vert x\\Vert_2/\\sqrt{n}} = \\sqrt{n}\\Vert A\\Vert_2\n",
+    "\\end{equation*}\n",
+    "\n",
+    "\n",
+    "Lets consider a vector x; \n",
+    "\\begin{equation}\n",
+    "    A\n",
+    "     =\n",
+    "    \\begin{bmatrix}\n",
+    "    1 & 3\\\\\n",
+    "    2 & 1\\\\\n",
+    "    3 & 4\n",
+    "    \\end{bmatrix}\n",
+    "\\end{equation}\n",
+    "\n",
+    "Here, \n",
+    "\\begin{equation*}\n",
+    "\\Vert A\\Vert_\\infty = max \\Vert a_{i}^{*}\\Vert _1 , where (1\\leq i \\leq m) = 7 \n",
+    "\\end{equation*}\n",
+    "\n",
+    "However, finding the 2-norm is a bit of a problem. \n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "## (d) $$||A||_2 \\leq \\sqrt{m}||A||_\\infty $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "collapsed": true,
+    "deletable": true,
+    "editable": true
+   },
+   "source": [
+    "\\begin{equation*}\n",
+    "\\Vert A\\Vert_2 = sup \\frac{\\Vert Ax\\Vert_2}{\\Vert x\\Vert_2}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "From the inequality b and $Ax$ is in $\\mathbb{C}^m$, we can write:\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "sup \\frac{\\Vert Ax\\Vert_2}{\\Vert x\\Vert_2} \\leq sup \\sqrt{m}\\frac{\\Vert Ax\\Vert_\\infty}{\\Vert x\\Vert_2}\n",
+    "\\end{equation*}\n",
+    "\n",
+    "And from the inequality a,\n",
+    "\n",
+    "\\begin{equation*}\n",
+    "sup \\sqrt{m}\\frac{\\Vert Ax\\Vert_\\infty}{\\Vert x\\Vert_2} \\leq sup \\sqrt{m}\\frac{\\Vert Ax\\Vert_\\infty}{\\Vert x \\Vert_\\infty} = \\sqrt{m} \\Vert A\\Vert_\\infty\n",
+    "\\end{equation*}"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "collapsed": true
+   },
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Octave",
+   "language": "octave",
+   "name": "octave"
+  },
+  "language_info": {
+   "file_extension": ".m",
+   "help_links": [
+    {
+     "text": "GNU Octave",
+     "url": "https://www.gnu.org/software/octave/support.html"
+    },
+    {
+     "text": "Octave Kernel",
+     "url": "https://github.com/Calysto/octave_kernel"
+    },
+    {
+     "text": "MetaKernel Magics",
+     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+    }
+   ],
+   "mimetype": "text/x-octave",
+   "name": "octave",
+   "version": "4.2.0"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}