# Ec securehostcomsiamot71html buy from amazoncom 624

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 616 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online