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 1 Diagonal Matrices 2 Triangular Matrices 3 Symmetric Matrices
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 . Since only the main diagonal entries of a diagonal matrix are significant, we may write a diagonal matrix as A = diag { a 11 , a 22 , . . . , a nn } .

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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 } .

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