67 s a gerschgorin illustrated the use of gerschgorin

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Unformatted text preview: ies of A. Consequently, the eigenvalues of A must vary continuously with the entries of A. Caution! Components of an eigenvector need not vary continuously with the entries of A —e.g., consider x = ( −1 , 1)T as an eigenvector for A = 0 1 , and let → 0. 0 D E Example 7.1.4 Spectral Radius. For square matrices A, the number T H ρ(A) = max |λ| λ∈σ (A) is called the spectral radius of A. It’s not uncommon for applications to require only a bound on the eigenvalues of A. That is, precise knowledge of each eigenvalue may not called for, but rather just an upper bound on ρ(A) is all that’s often needed. A rather crude (but cheap) upper bound on ρ(A) is obtained by observing that ρ(A) ≤ A for every matrix norm. This is true because if (λ, x) is any eigenpair, then X = x | 0 | · · · | 0 n×n = 0, and λX = AX implies |λ| X = λX = AX ≤ A X , so Y P IG R |λ| ≤ A for all λ ∈ σ (A) . (7.1.12) This result is a precursor to a stronger relationship between spectral radius and norm that is hinted at in Exercise 7.3.12 and developed in Example 7.10.1 (p. 619). O C The eigenvalue bound (7.1.12) given in Example 7.1.4 is cheap to compute, especially if the 1-norm or ∞ -norm is used, but you often get what you pay for. You get one big circle whose radius is usually much larger than the spectral 67 radius ρ(A). It’s possible to do better by using a set of Gerschgorin circles as described below. 67 S. A. Gerschgorin illustrated the use of Gerschgorin circles for estimating eigenvalues in 1931, but the concept appears earlier in work by L. L´vy in 1881, by H. Minkowski (p. 278) in 1900, e and by J. Hadamard (p. 469) in 1903. However, each time the idea surfaced, it gained little attention and was quickly forgotten until Olga Taussky (1906–1995), the premier woman of linear algebra, and her fellow German emigr` Alfred Brauer (1894–1985) became captivated e by the result. Taussky (who became Olga Taussky-Todd after marrying the numerical analyst John Todd) and Br...
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