Chapter 1.6 SpecialMatrices

Chapter 1.6 SpecialMatrices - Outline Diagonal Matrices...

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Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Special Matrices Math 2270 K. J. Platt, Ph.D. Department of Mathematics Snow College Spring 2014

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Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Outline
Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Diagonal Matrices Deﬁnition Let A = [ a ij ] be a square matrix. A is a diagonal matrix if a ij = 0 whenever i 6 = j .

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Outline Diagonal Matrices Triangular Matrices Symmetric Matrices Properties of Diagonal Matrices Theorem Suppose A = diag { a 11 , a 22 , ... , a nn } and r is a non-negative integer. (a) A r = diag { a r 11 , a r 22 a r nn } . (b) A is invertible if and only if a ii 6 = 0 for all i, and in this case A - r = diag { a - r 11 , a - r 22 a - r nn } . 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 { d 1 , d 2 , ... , d n } . (a) If B is an n × m matrix with row vectors w 1 , w 2 ,..., w n , then AB = d 1 w 1 d 2 w 2 .

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Chapter 1.6 SpecialMatrices - Outline Diagonal Matrices...

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