**Unformatted text preview: **ize of the
largest Jordan block associated with an eigenvalue λ is k (i.e., if index (λ) = k ),
then f (λ), f (λ), . . . , f (k−1) (λ) must exist in order for f (J) to make sense. D
E Matrix Functions For A ∈ C n×n with σ (A) = {λ1 , λ2 , . . . , λs } , let ki = index (λi ).
• • T
H A function f : C → C is said to be deﬁned (or to exist) at A when
f (λi ), f (λi ), . . . , f (ki −1) (λi ) exist for each λi ∈ σ (A) .
.. Suppose that A = PJP−1 , where J = .J IG
R .. is in Jordan form . with the J ’s representing the various Jordan blocks described on
p. 590. If f exists at A, then the value of f at A is deﬁned to be
⎛ f (A) = Pf (J)P−1 = P ⎝ Y
P .. . ⎞ f (J ) ⎠ P−1 ,
..
. (7.9.3) where the f (J ) ’s are as deﬁned in (7.9.2). O
C We still need to explain why (7.9.3) produces a uniquely deﬁned matrix.
The following argument will not only accomplish this purpose, but it will also
establish an alternate expression for f (A) that involves neither the Jordan form
J nor the transforming matrix P. Begin by partitioning J into its s Jordan
segments as described on p. 590, and partition P and P−1 conformably as P = P1 | · · · | Ps , ⎛
J=⎝ J(λ1 ) .. ⎠, .
J(λs ) ⎞
Q1
⎜.⎟
= ⎝ . ⎠.
.
⎛ ⎞
and P−1 Qs Deﬁne Gi = Pi Qi , and observe that if ki = index (λi ), then Gi is the projector onto N (A − λi I)ki along R (A − λi I)ki . To see this, notice that
Li = J(λi ) − λi I is nilpotent of index ki , but J(λj ) − λi I is nonsingular when Copyright c 2000 SIAM Buy online from SIAM
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602
Chapter 7
Eigenvalues and Eigenvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540
i = j, so (A − λi I) = P(J − λi I)P−1 ⎞ ⎛ J (λ ) − λ I
1
i
..
⎜
.
⎜
Li
= P⎜
⎝
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Please report violations to [email protected] .
J(λs ) − λi I ⎟
⎟ −1
⎟P
⎠ (7.9.4) is a core-nilpo...

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