Since p q log10 1 log10 h i k i k we see

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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 . ji Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 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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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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