**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
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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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