pset 3

psets/pset2.ipynb

@@ -0,0 +1,63 @@
+{
+ "cells": [
+  {
+   "cell_type": "markdown",
+   "id": "0f2a9965",
+   "metadata": {},
+   "source": [
+    "# 18.085 Problem Set 2\n",
+    "\n",
+    "Due Friday **March 3** at 1pm."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "d9e8ae37",
+   "metadata": {},
+   "source": [
+    "## Problem 1\n",
+    "\n",
+    "**(a)** The eigenvalues of a real *anti-Hermitian* matrix $A = -A^T$ must be \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_.   Derive this by considering $\\overline{x^T} A x$ for an eigenvector $Ax = \\lambda x$, and demonstrate it numerically by constructing a random anti-Hermitian matrix `A = randn(n,n); A = A - A'` in Julia and looking at its eigenvalues `using LinearAlgebra; eigvals(A)` for `n=4` and `n=5`.  (You might want to try the numerical experiment first if you aren't sure what the answer is.)\n",
+    "\n",
+    "**(b)** Suppose $A$ is a $3 \\times 3$ real-symmetric matrix with eigenvalues $\\lambda_1 = -2$, $\\lambda_2 = 3$, and $\\lambda_3 = -1$, with corresponding orthonormal eigenvectors $q_1, q_2, q_3$.  In terms of these quantities, give the (full) SVD of $A$.\n",
+    "\n",
+    "**(c)** Construct a random $5 \\times 3$ matrix `A = randn(5, 3)` and form a related *real-symmetric* matrix `B = [ 0I A; A' 0I ]`, corresponding to\n",
+    "$$\n",
+    "B = \\begin{pmatrix} 0 & A \\\\ A^T & 0 \\end{pmatrix} .\n",
+    "$$\n",
+    "Compare the eigenvalues of $B$ (`eigvals(B)`) to the singular values of $A$ (`svdvals(A)`).  What do you notice?  Explain it by using the SVD $A = U \\Sigma V^T$ to construct eigenvalues and eigenvectors of $B$."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "88ac965b",
+   "metadata": {},
+   "source": [
+    "## Problems 2, 3, etc: coming soon"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "e1fca19a",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Julia 1.8.0",
+   "language": "julia",
+   "name": "julia-1.8"
+  },
+  "language_info": {
+   "file_extension": ".jl",
+   "mimetype": "application/julia",
+   "name": "julia",
+   "version": "1.8.2"
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+ "nbformat_minor": 5
+}