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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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.
 Spring '10
 Sophie
 Linear Algebra, Algebra, Matrices

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