This situation is depicted by drawing a transition

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Unformatted text preview: nonsingular is 1 ∈ σ (A) , while / convergence of the Neumann series requires each |λ| < 1. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 528 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Example 7.3.2 Eigenvalue Perturbations. It’s often important to understand how the eigenvalues of a matrix are affected by perturbations. In general, this is a complicated issue, but for diagonalizable matrices the problem is more tractable. Problem: Suppose B = A + E, where A is diagonalizable, and let β ∈ σ (B) . If P−1 AP = D = diag (λ1 , λ2 , . . . , λn ) , explain why It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu min |β − λi | ≤ κ(P) E , λi ∈σ (A) where κ(P) = P P−1 (7.3.10) D E for matrix norms satisfying D = maxi |λi | (e.g., any standard induced norm). Solution: Assume β ∈ σ (A) —(7.3.10) is trivial if β ∈ σ (A) —and observe that T H (β I − A)−1 (β I − B) = (β I − A)−1 (β I − A − E) = I − (β I − A)−1 E implies that 1 ≤ (β I − A)−1 E —otherwise I − (β I − A)−1 E is nonsingular by (7.3.9), which is impossible because (β I − B) (and hence (β I − A)−1 (β I − B) is singular). Consequently, IG 1 ≤ (β I − A)−1 E = P(β I − D)−1 P−1 E ≤ P (β I − D)−1 1 = κ(P) E max |β − λi |−1 = κ(P) E , i mini |β − λi | P−1 E R Y and this produces (7.3.10). Similar to the case of linear systems (Example 5.12.1, p. 414), the expression κ(P) is a condition number in the sense that if κ(P) is relatively small, then the λi ’s are relatively insensitive, but if κ(P) is relatively large, we must be suspicious. Note: Because it’s a corollary of their 1960 results, the bound (7.3.10) is often referred to as the Bauer–Fike bound . P O C Infinite series representations can always be avoided because every function of An×n can be expressed as...
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