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Unformatted text preview: Revisited. The Cayley–Hamilton theorem (p. 509) says
that if p(λ) = 0 is the characteristic equation for A, then p(A) = 0. This is
evident for diagonalizable A because p(λi ) = 0 for each λi ∈ σ (A) , so, by
(7.3.6), p(A) = p(λ1 )G1 + p(λ2 )G2 + · · · + p(λk )Gk = 0.
Problem: Establish the Cayley–Hamilton theorem for nondiagonalizable matrices by using the diagonalizable result together with a continuity argument.
Solution: Schur’s triangularization theorem (p. 508) insures An×n = UTU∗
for a unitary U and an upper triangular T having the eigenvalues of A on the Copyright c 2000 SIAM Buy online from SIAM
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7.3 Functions of Diagonalizable Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540 533 diagonal. For each = 0, it’s possible to ﬁnd numbers i such that (λ1 +
(λ2 + 2 ), . . . , (λn + n ) are distinct and
i = | |. Set
D( ) = diag ( 1 , 2, . . . , n) and 1 ), B( ) = U T + D( ) U∗ = A + E( ), It is illegal to print, duplicate, or distribute this material
Please report violations to firstname.lastname@example.org where E( ) = UD( )U∗ . The (λi + i ) ’s are the eigenvalues of B( ) and
they are distinct, so B( ) is diagonalizable—by (7.2.6). Consequently, B( )
satisﬁes its own characteristic equation 0 = p (λ) = det (A + E( ) − λI) for
each = 0. The coeﬃcients of p (λ) are continuous functions of the entries in
E( ) (recall (7.1.6)) and hence are continuous functions of the i ’s. Combine
this with lim →0 E( ) = 0 to obtain 0 = lim →0 p (B( )) = p(A). D
E Note: Embedded in the above development is the fact that every square complex matrix is arbitrarily close to some diagonalizable matrix because for each
= 0, we have A − B( ) F = E( ) F = (recall Exercise 5.6.9). T
H Example 7.3.7
74 Power method is an iterative technique for computing a dominant eigenpair
(λ1 , x) of a diagonalizable A ∈ m×m with eigenvalues IG
R |λ1 | > |λ2 | ≥ |λ3 | ≥ · · · ≥ |λk |. Note that this implies λ1 is real—ot...
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