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Unformatted text preview: . That is, let h(z ) = f (g (z )), and use Exercise 7.9.17 to prove that if g (J ) and f g (J ) each exist, then ⎛ ⎞ λ10 h(J ) = f g (J ) for J = ⎝ 0 λ 1 ⎠ . 00λ D E 7.9.19. Prove that if Γi is a simple closed contour enclosing λi ∈ σ (A) but excluding all other eigenvalues of A, then the ith spectral projector is 1 1 Gi = (ξ I − A)−1 dξ = R(ξ )dξ. 2π i Γi 2π i Γi 7.9.20. For f (z ) = z −1 , verify that f (A) = A −1 T H for every nonsingular A. IG R 7.9.21. If Γ is a simple closed contour enclosing all eigenvalues of a nonsingular 1 matrix A, what is the value of ξ −1 (ξ I − A)−1 dξ ? 2π i Γ Y P 7.9.22. Generalized Inverses. The inverse function f (z ) = z −1 is not defined at singular matrices, but the generalized inverse function z −1 if z = 0, 0 if z = 0, is defined on all square matrices. It’s clear from Exercise 7.9.20 that if A is nonsingular, then g (A) = A−1 , so g (A) is a natural way to extend the concept of inversion to include singular matrices. Explain why g (A) = AD is the Drazin inverse of Example 5.10.5 (p. 399) and not necessarily the Moore–Penrose pseudoinverse A† described on p. 423. g (z ) = O C 7.9.23. Drazin Is “Natural.” Suppose that A is a singular matrix, and let Γ be a simple closed contour that contains all eigenvalues of A except λ1 = 0, which is neither in nor on Γ. Prove that 1 ξ −1 (ξ I − A)−1 dξ = AD 2π i Γ is the Drazin inverse for A as defined in Example 5.10.5 (p. 399). Hint: The Cauchy–Goursat theorem states that if a function f is analytic at all points inside and on a simple closed contour Γ, then Γ f (z )dz = 0. Copyright c 2000 SIAM Buy online from SIAM Buy from 616 Chapter 7 Eigenvalues and Eigenvectors 7.10 DIFFERENCE EQUATIONS, LIMITS, AND SUMMABILITY A linear difference equation of order m with constant coefficients has the form It is illegal to print, duplicate, or distribute this material Please repor...
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