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**Unformatted text preview: **genvectors
http://www.amazon.com/exec/obidos/ASIN/0898714540 It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu • In other words, if Γ is a simple closed contour containing σ (A) in its
interior, then
1
f (A) =
f (ξ )(ξ I − A)−1 dξ
(7.9.22)
2π i Γ
whenever f is analytic in and on Γ. Since this formula makes sense for
general linear operators, it is often adopted as a deﬁnition for f (A) in more
general settings. • Furthermore, if Γi is a simple closed contour enclosing λi but excluding all
other eigenvalues of A, then the ith spectral projector is given by
1
2π i Gi = R(ξ )dξ =
Γi 1
2π i (ξ I − A)−1 dξ D
E (Exercise 7.9.19). Γi T
H Exercises for section 7.9 7.9.1. Lake #i in a closed system of three lakes of equal volume V initially
contains ci lbs of a pollutant. If the water in the system is circulated
at rates (gal/sec) as indicated in Figure 7.9.2, ﬁnd the amount of pollutant in each lake at time t > 0 (assume continuous mixing), and then
determine the pollution in each lake in the long run. IG
R
2r Y
P
4r #1 3r #2 2r O
C #3 r Figure 7.9.2 7.9.2. Suppose that A ∈ C n×n has eigenvalues λi with index (λi ) = ki . Explain why the ith spectral projector is given by
Gi = fi (A), where fi (z ) = 1 when z = λi ,
0 otherwise. 7.9.3. Explain why each spectral projector Gi can be expressed as a polynomial in A.
7.9.4. If σ (An×n ) = {λ1 , λ2 , . . . , λs } with ki = index (λi ), explain why
s ki −1 Ak =
i=1 j =0 Copyright c 2000 SIAM k k−j
λ (A − λi I)j Gi .
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7.9 Functions of Nondiagonalizable Matrices
http://www.amazon.com/exec/obidos/ASIN/0898714540
k
j 7.9.5. With the convention that
⎛
⎛ It is illegal to print, duplicate, or distribute this material
Please report violations to meyer@ncsu.edu ⎝ λ ⎞k 1
.. ... ⎠
.
λ m×m λk ⎜
⎜
⎜
⎜
⎜
=⎜
⎜
⎜
⎜
⎜
⎝ 613 = 0 for j > k,...

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