Chapter 1.6 SpecialMatrices

A is symmetric if and only if at a in particular aij

Info iconThis preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: atrix. A is symmetric if and only if AT = A. In particular, aij = aji for all i , j . Example Which matrices are symmetric? 1 2 −3 4 , A= 2 3 −3 4 5 1 00 B = 0 −2 0 0 03 1 2 −3 0 C = −2 −2 3 0 3 Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Properties of Symmetric Matrices Theorem Suppose A = [aij ] and B = [bij ] are symmetric n × n matrices, and λ is a scalar. (a) AT is symmetric. (b) A + B is symmetric. (c) λA is symmetric. (d) AB is symmetric if and only if A and B commute. (e) If A is invertible, then A−1 is symmetric. Outline Diagonal Matrices Triangular Matrices Symmetric Matrices The Proof Proof. (a) (AT )T = A = AT . (b) (A + B )T = AT + B T = A + B . (c) (λA)T = λAT = λA. (d) (AB )T = B T AT = BA, so AB is symmetric if and only if A and B commute. (e) If A is invertible, then so is AT and (A−1...
View Full Document

This note was uploaded on 02/10/2014 for the course MATH 2270 taught by Professor Kenyonj.platt during the Spring '14 term at Snow College.

Ask a homework question - tutors are online