M235A8 - MATH 235 Assignment 8 Orthogonal Diagonalization,...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MATH 235 Assignment 8 Orthogonal Diagonalization, Symmetric & Complex Matrices Hand in questions 1,3,5,7,9,11 by 9:30 am on Wednesday November 22, 2006. 1. (a) Let A,B be n n matrices and suppose B = A- 1 A T and B is symmetric. Prove that A 2 is symmetric. (b) Suppose C is a real n n matrix such that C is symmetric and C 2 = C and let D = I n- 2 C with I n denoting the n n identity matrix. Prove that D is symmetric and orthogonal. 2. (a) Find all complex 2 2 matrices A = [ a ij ] which are both unitary and Hermitian, and have a 11 = 1 / 2. (b) Show that a real 2 2 normal matrix is either symmetric or has the form " a b- b a # ( a,b R ). 3. Let Z be an m n complex matrix such that Z * Z = I n where I n denotes the n n identity matrix. (a) Show that H = ZZ * is Hermitian and satisfies H 2 = H . (b) Show that U = I n- 2 ZZ * is both unitary and Hermitian. 4. Let A be a complex n n matrix. (a) Show that A = H + K for some Hermitian matrix H and some skew-Hermitian matrix K ....
View Full Document

This note was uploaded on 01/05/2012 for the course MATH 235 taught by Professor Celmin during the Fall '08 term at Waterloo.

Page1 / 3

M235A8 - MATH 235 Assignment 8 Orthogonal Diagonalization,...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online