Chapter 1.6 SpecialMatrices

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

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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...
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## 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.

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