Chapter 1.6 SpecialMatrices

ann b a is invertible if and only if aii 0 for

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Unformatted text preview: s invertible if and only if aii = 0 for all i, and in this case − − − A−r = diag{a11r , a22r , . . . , annr }. In particular, products and inverses of diagonal matrices are diagonal. Proof. Induction. Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Products with Diagonal Matrices Theorem Suppose A = diag{d1 , d2 , . . . , dn }. (a) If B is an n × m matrix with row vectors w1 , w2 , . . . , wn , then d1 w1 d2 w2 AB = . . . dn wn (b) If B is an m × n matrix with column vectors v1 , v2 , . . . , vn , then BA = d1 v1 d2 v2 · · · dn vn Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Examples Example 1 2 3 Discuss the invertibility of the diagonal 1 0 0 0 , B = A = 0 −2 0 0 −3 matrices 1 0 0 0 0 0 0 0 0 0 0 0 5 0 0 −2 Find B 2 , A−1 , and A−3 for A and B in Example 1. Demonstrate the pr...
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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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