MA 3065 Study Guide - Final Guide: Rayleigh Quotient, Symmetric Matrix, Ellipse

• School: Minneapolis College of Art and Design
• Department: Filmmaking-Sound
• Course: MA 3065
• Professor: p

Document Summary

Math 5588 homework 7 (due thursday march 9) Recall a matrix a = (aij) rn n is symmetric if aij = aji, so that at = a. We say a symmetric matrix a is positive de nite, written a 0, if vt av = nxi=1 nxj=1 aijvivj 0 for all v rn. We write a b whenever b a 0: a real symmetric matrix a rn n has n real eigenvalues 1 2 n, counted with multiplicity. Show that a is positive de nite (a 0) if and only if 1 0. 1 = min v6=0 vt av vt v. The right hand side above is called a rayleigh quotient. : let a r2 2 be a symmetric matrix. Show that a is positive de nite (a 0) if and only if det(a) 0 and trace(a) 0. 3. (a) let a, b rn n be diagonal matrices.