T h a crude estimate is derived from the bound given

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Unformatted text preview: (7.1.14), let D = diag (a11 , a22 , . . . , ann ) and B = A − D, and set C(t) = D + tB for t ∈ [0, 1]. The first part shows that the eigenvalues of λi (t) of C(t) are contained in the union of the Gerschgorin circles Ci (t) defined by |z − aii | ≤ t ri . The circles Ci (t) grow continuously with t from individual points aii when t = 0 to the Gerschgorin circles of A when t = 1, Copyright c 2000 SIAM Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu Buy from AMAZON.com 7.1 Elementary Properties of Eigensystems http://www.amazon.com/exec/obidos/ASIN/0898714540 499 so, if the circles in the isolated union U are centered at ai1 i1 , ai2 i2 , . . . , aik ik , then for every t ∈ [0, 1] the union U (t) = Ci1 (t) ∪ Ci2 (t) ∪ · · · ∪ Cik (t) is disjoint from the union U (t) of the other n − k Gerschgorin circles of C(t). Since (as mentioned in Example 7.1.3) each eigenvalue λi (t) of C(t) also varies continuously with t, each λi (t) is on a continuous curve Γi having one end at λi (0) = aii and the other end at λi (1) ∈ σ (A) . But since U (t) ∩ U (t) = φ for all t ∈ [0, 1], the curves Γi1 , Γi2 , . . . , Γik are entirely contained in U , and hence the end points λi1 (1), λi2 (1), . . . , λik (1) are in U . Similarly, the other n − k eigenvalues of A are in the union of the complementary set of circles. D E Example 7.1.5 5 0 1 Problem: Estimate the eigenvalues of A = 1 6 0 1 1 −5 . T H • A crude estimate is derived from the bound given in Example 7.1.4 on p. 497. Using the ∞ -norm, (7.1.12) says that |λ| ≤ A ∞ = 7 for all λ ∈ σ (A) . • Better estimates are produced by the Gerschgorin circles in Figure 7.1.2 that are derived from row sums. Statements (7.1.13) and (7.1.14) guarantee that one eigenvalue is in (or on) the circle centered at −5, while the remaining two eigenvalues are in (or on) the larger circle centered at +5. Y P -7 Figure 7.1.2. • IG R -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 Gerschgorin circles derived from row sums. The best estimate is obtained from (7.1.16) by considering Gr ∩ Gc . O C -7 Figure 7.1.3...
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