**Unformatted text preview: **nonsingular is 1 ∈ σ (A) , while
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convergence of the Neumann series requires each |λ| < 1. Copyright c 2000 SIAM Buy online from SIAM
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528
Chapter 7
Eigenvalues and Eigenvectors
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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 [email protected] 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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