113 and 7115 means that the eigenvalues of a are

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Unformatted text preview: auer devoted significant effort to strengthening, promoting, and popularizing Gerschgorin-type eigenvalue bounds. Their work during the 1940s and 1950s ended the periodic rediscoveries, and they made Gerschgorin (who might otherwise have been forgotten) famous. Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 498 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Gerschgorin Circles The eigenvalues of A ∈ C n×n are contained in the union Gr of the n Gerschgorin circles defined by • It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] n |z − aii | ≤ ri , where ri = |aij | for i = 1, 2, . . . , n. (7.1.13) D E j =1 j =i In other words, the eigenvalues are trapped in the collection of circles centered at aii with radii given by the sum of absolute values in Ai∗ with aii deleted. T H • Furthermore, if a union U of k Gerschgorin circles does not touch any of the other n − k circles, then there are exactly k eigenvalues (counting multiplicities) in the circles in U . (7.1.14) • Since σ (AT ) = σ (A) , the deleted absolute row sums in (7.1.13) can be replaced by deleted absolute column sums, so the eigenvalues of A are also contained in the union Gc of the circles defined by IG R n |z − ajj | ≤ cj , where cj = |aij | for j = 1, 2, . . . , n. Y P • (7.1.15) i=1 i=j Combining (7.1.13) and (7.1.15) means that the eigenvalues of A are contained in the intersection Gr ∩ Gc . (7.1.16) Proof. Let (λ, x) be an eigenpair for A, and assume x has been normalized so that x ∞ = 1. If xi is a component of x such that |xi | = 1, then O C n n λxi = [λx]i = [Ax]i = aij xj =⇒ (λ − aii )xi = j =1 and hence |λ − aii | =|λ − aii | |xi | = aij xj ≤ j =i aij xj , j =1 j =i |aij | |xj | ≤ j =i |aij | = ri . j =i Thus λ is in one of the Gerschgorin circles, so the union of all such circles contains σ (A) . To establish...
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