Lecture21 - 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 Winter 2008 Lecture 21 Diagonalizable Matrices Sophie Chrysostomou definition 0.1 Let A be an n × n matrix with characteristic polynomial p(λ) and distinct eigenvalues λ1 , λ2 , · · · , λr . Suppose that (λ − λi )αi is a factor of p(λ), but (λ − λi )αi +1 is not, then we say that αi is the algebraic multiplicity of λi . If dim(Eλi = βi , then we say that βi is the geometric multiplicity of λi . theorem 0.2 The geometric multiplicity of an eigenvalue of an n × n matrix A is less than or equal to its algebraic multiplicity. 31 example 0.3 Let A = 0 3 00 each eigenvalue find its algebraic 0 1 . Find all the eigenvalues of A. For 5 and geometric multiplicity. 1 theorem 0.4 Let A be an n × n matrix with a characteristic polynomial p(λ) = (λ − λ1 )α1 (λ − λ2 )α2 · · · (λ − λm )αm . Then A is diagonalizable ⇐⇒ algebraic multiplicity of λi = geometric multiplicity of λi for all i = 1, 2, · · · , m. 2 3 −3 −3 3 −3 . Determine if A is diagonalizable. example 0.5 Let A = −3 −3 −3 3 If it is, find am invertible matrix P , and a diagonal matrix D such that P −1 AP = D. 3 5 −2 0 0 −2 5 0 0 . Determine if A is diagonalizexample 0.6 Let A = 0 0 7 4 0 047 able. If it is, find am invertible matrix P , and a diagonal matrix D such that P −1 AP = D. 4 ...
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