Lecture20 - University of Toronto at Scarborough Department...

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Unformatted text preview: University of Toronto at Scarborough Department of Computer & Mathematical Sciences MAT A23 Lecture 20 Diagonalizable Matrices Sophie Chrysostomou definition 0.1 Let A and D be an n × n matrices. a) D is a diagonal matrix, if all its entries not on the main diagonal are zero. ie. dij = 0 for all i = j . b) A is diagonalizable if there exists an invertible n × n matrix P such that P −1 AP is a diagonal matrix. theorem 0.2 Let A be an n × n matrix. A is diagonalizable ⇐⇒ A has n linearly independent eigenvectors. Winter 2008 1 . 2 theorem 0.3 Let v1 , v2 , · · · , vk be eigenvectors corresponding to the distinct eigenvalues λ1 , λ2 , · · · , λk of the square matrix A, then v1 , v2 , · · · , vk are linearly independent. 3 corollary 0.4 If A is n × n and has n distinct eigenvalues, then A is diagonalizable. 3 0 example 0.5 Let A = 7 is, find the invertible matrix 0 5 0 P 7 0 . Determine if A is diagonalizable. If it 3 so that P −1 AP is diagonalizable. 4 ...
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This note was uploaded on 05/22/2010 for the course MATH MATA23 taught by Professor Sophie during the Spring '10 term at University of Toronto.

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