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"# MATH 382 - Fall 2018\n",
"## Scientific Computing\n",
"Yor Name Here\n",
"\n",
"${\\bf Directions:}$ Write the solutions of the five problems below (code and text as needed) in this notebook and submit it in canvas.\n",
"
\n",
"# Lab Assignment 1: Linear Algebra\n",
"
"
]
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"## Problem \\#1\n",
"\n",
"Create the following matrices and print them along with their data type\n",
"\n",
"(a) A $3 \\times 3$ diagonal matrix $A$ of type double with diagonal entries $(1, -1, 2)$"
]
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"name": "stdout",
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"text": [
"A = [[ 1. 0. 0.]\n",
" [ 0. -1. 0.]\n",
" [ 0. 0. 2.]] \n",
"\n",
"float64 \n",
"\n"
]
}
],
"source": [
"# Enter your code here"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"(b) A $4 \\times 3$ diagonal matrix $B$ of type int with diagonal entries $(1, -1, 2)$. $Hint:$ Use np.vstack"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
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{
"name": "stdout",
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"text": [
"B = [[ 1 0 0]\n",
" [ 0 -1 0]\n",
" [ 0 0 2]\n",
" [ 0 0 0]] \n",
"\n",
"int64 \n",
"\n"
]
}
],
"source": [
"# Enter your code here"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"(c) A $3 \\times 4$ matrix $C$ of type double with diagonal entries $(1, -1, 2)$. $Hint:$ Use np.hstack"
]
},
{
"cell_type": "code",
"execution_count": 31,
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{
"name": "stdout",
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"text": [
"C = [[ 1. 0. 0. 0.]\n",
" [ 0. -1. 0. 0.]\n",
" [ 0. 0. 2. 0.]] \n",
"\n",
"float64 \n",
"\n"
]
}
],
"source": [
"# Enter your code here"
]
},
{
"cell_type": "markdown",
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"deletable": true,
"editable": true
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"source": [
"
\n",
"## Problem \\#2\n",
"(a) Create the $4 \\times 4$ matrix of type double\n",
"$$\n",
"A = \\left(\\begin{array}{cccc} 2 & -1 & 0 & 0 \\\\ -1 & 2 & -1 & 0\\\\ 0 & -1 & 2 & -1 \\\\ 0 & 0 & -1 & 2 \\end{array} \\right)\n",
"$$\n",
"And print it along with its data type"
]
},
{
"cell_type": "code",
"execution_count": 61,
"metadata": {
"collapsed": false,
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"name": "stdout",
"output_type": "stream",
"text": [
"A = [[ 2. -1. 0. 0.]\n",
" [-1. 2. -1. 0.]\n",
" [ 0. -1. 2. -1.]\n",
" [ 0. 0. -1. 2.]] \n",
"\n",
"float64 \n",
"\n"
]
}
],
"source": [
"# Enter your code here"
]
},
{
"cell_type": "markdown",
"metadata": {
"deletable": true,
"editable": true
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"source": [
"
\n",
"\n",
"(b) Is A invertable? Explain your answer"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"(c) Find the eigen decomposition of $A$, $A = V \\Lambda V^{-1}$, and print the matrices $V$ and $\\Lambda$"
]
},
{
"cell_type": "code",
"execution_count": 62,
"metadata": {
"collapsed": false,
"deletable": true,
"editable": true
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"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"V = [[-0.37174803 -0.60150096 -0.37174803 -0.60150096]\n",
" [ 0.60150096 0.37174803 -0.60150096 -0.37174803]\n",
" [-0.60150096 0.37174803 -0.60150096 0.37174803]\n",
" [ 0.37174803 -0.60150096 -0.37174803 0.60150096]] \n",
"\n",
"Lambda = [[ 3.61803399 0. 0. 0. ]\n",
" [ 0. 2.61803399 0. 0. ]\n",
" [ 0. 0. 0.38196601 0. ]\n",
" [ 0. 0. 0. 1.38196601]] \n",
"\n"
]
}
],
"source": [
"# Enter your code here"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"\n",
"(d) Is the matrix $V$ orthogonal? Explain your answer"
]
},
{
"cell_type": "code",
"execution_count": null,
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"collapsed": true
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"
\n",
"## Problem \\#3\n",
"\n",
"Consider the syste $A{\\bf x} = {\\bf b}$ with\n",
"$$\n",
"A = \\left( \\begin{array}{ccc} 1 & -1 & 2 \\\\ 2 & 0 & -1 \\\\ 1 & 0 & 2 \\\\ -2 & 3 & -2 \\end{array}\\right) \\ \\ \\ \\ \\ \\ \\ \\ \\ \\text{and} \\ \\ \\ \\ \\ \\ \\ \\ \\\n",
"{\\bf b} = \\left( \\begin{array}{ccc} 1 \\\\ 0 \\\\ 2 \\\\ 1 \\end{array}\\right)\n",
"$$\n",
"\n",
"(a) Does the system have any solutions? Explain\n",
"
\n",
"\n",
"(b) Find the Singular Value Decomposition of $A$, $A = U D V^\\top$, and print $U$, $D$, and $V$\n",
"
\n",
"\n",
"(c) Find and print $A$'s Moore-Penrose pseudo inverse $A^{+}$\n",
"
\n",
"\n",
"(d) Find and print the approximate solution to the given system, $\\widehat{\\bf x}$, that minimizes the error $\\| A{\\bf \\widehat{x}} - {\\bf b} \\|_2$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"
\n",
"## Problem \\#4\n",
"\n",
"Consider the system in Problem \\#3 with ${\\bf b} = \\left(1, 0, 2, \\alpha \\right)^\\top$\n",
"\n",
"(a) For what value(s) of $\\alpha$ does the system have a unique solution?\n",
"
\n",
"\n",
"(b) Find it."
]
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"
\n",
"## Problem \\#5\n",
"\n",
"For the matrix $A$ in Problem \\#3\n",
"\n",
"(a) Find the eigen value decomposition of the matrix $A^\\top A$.\n",
"
\n",
"\n",
"(b) How are the eigen values and eigen vectors of $A^\\top A$ related to the singular values and singular vectors of $A$?\n",
"
\n",
"\n",
"(c) Find the eigen value decomposition of the matrix $AA^\\top$. \n",
"
\n",
"\n",
"(d) How are the eigen values and eigen vectors of $AA^\\top$ related to the singular values and singular vectors of $A$?"
]
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