Unformatted text preview: explain why
1 λk−1 k
1 λk λk−2 · · ·
λk−1 .. ..
.. . . . λk k
m−1 λk−m+1 λk−2 k
1 λk−1 D
E λk 7.9.6. Determine e A for A = 6
H . √
7.9.7. For f (z ) = 4 z − 1, determine f (A) when A = 7.9.8. IG
−1 ⎞ −8
1 . (a) Explain why every nonsingular A ∈ C n×n has a square root.
(b) Give necessary and suﬃcient conditions for the existence of A
when A is singular. Y
P 7.9.9. Spectral Mapping Property. Prove that if (λ, x) is an eigenpair
for A, then (f (λ), x) is an eigenpair for f (A) whenever f (A) exists.
Does it also follow that alg multA (λ) = alg multf (A) (f (λ))? O
C 7.9.10. Let f be deﬁned at A, and let λ ∈ σ (A) . Give an example or an
explanation of why the following statements are not necessarily true.
(a) f (A) is similar to A.
(b) geo multA (λ) = geo multf (A) (f (λ)) .
(c) indexA (λ) = indexf (A) (f (λ)).
7.9.11. Explain why Af (A) = f (A)A whenever f (A) exists.
7.9.12. Explain why a function f is deﬁned at A ∈ C n×n if and only if f
is deﬁned at AT , and then prove that f (AT ) = f (A) . Why can’t
( ) be used in place of ( ) ? Copyright c 2000 SIAM Buy online from SIAM
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Eigenvalues and Eigenvectors
7.9.13. Use the technique of Example 7.9.5 (p. 608) to establish the following
(a) eA e−A = I for all A ∈ C n×n .
for all α ∈ C and A ∈ C n×n .
(b) eαA = eA
(c) e = cos A + i sin A for all A ∈ C n×n . It is illegal to print, duplicate, or distribute this material
Please report violations to [email protected] 7.9.14. (a) Show that if AB = BA, then eA+B = eA eB .
(b) Give an example to show that eA+B = eA eB in general. D
E 7.9.15. Find the Hermite interpolation polynomial p(z ) as described in Example 7.9.4 such that p(A) = eA for A = 3
−1 . T
H 7.9.16. The Cayley–Hamilton theorem (pp. 509, 532) says that every A ∈ C n×n
satisﬁes its own character...
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