Edu for a c mn with rank a r the following statements

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Unformatted text preview: http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.5 Normal Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 551 ⊥ ˜ But this inequality is reversible because if V = {e1 , e2 , . . . , ei−1 } , then every ˜ y ∈ V has the form y = (0, . . . , 0, yi , . . . , yn )T , and hence n y∗ Dy = n λj |yj |2 ≤ λi j =i |yj |2 = λi for all y ∈ SV . ˜ j =i So min max y∗ Dy ≤ max y∗ Dy ≤ λi , and thus min max y∗ Dy = λi . It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] V SV SV ˜ V SV The value of the Courant–Fischer theorem is its ability to produce inequalities concerning eigenvalues of hermitian matrices without involving the associated eigenvectors. This is illustrated in the following two important examples. D E Example 7.5.2 Eigenvalue Perturbations. Let λ1 ≥ λ2 ≥ · · · ≥ λn be the eigenvalues of a hermitian A ∈ C n×n , and suppose A is perturbed by a hermitian E with eigenvalues 1 ≥ 2 ≥ · · · ≥ n to produce B = A + E, which is also hermitian. T H Problem: If β1 ≥ β2 ≥ · · · ≥ βn are the eigenvalues of B, explain why λi + 1 IG R ≥ βi ≥ λi + n for each i. (7.5.6) Solution: If U is a unitary matrix such that U∗ AU = D = diag (λ1 , . . . , λn ), ˜ ˜ then B = U∗ BU and E = U∗ EU have the same eigenvalues as B and E, ˜ ˜ respectively, and B = D + E. For x ∈ F = span {e1 , e2 , . . . , ei } with x 2 = 1, Y P x = (x1 , . . . , xi , 0, . . . , 0)T O C i i and x∗ Dx = λj |xj |2 ≥ λi j =1 |xj |2 = λi , j =1 ˜ so applying the max-min part of the Courant–Fischer theorem to B yields βi = max dim V =i ˜ ˜ min x∗ Bx ≥ min x∗ Bx = min x∈V x 2 =1 x∈F x 2 =1 x∈F x 2 =1 ˜ x∗ Dx + x∗ Ex ˜ ˜ ≥ min x∗ Dx + min x∗ Ex ≥ λi + min x∗ Ex = λi + n x∈F x 2 =1 x∈F x 2 =1 x∈C x 2 =1 n, where the last equality is the result of the “min” part of (7.5.4). Similarly, for x ∈ T = span {ei , . . . , en } with x 2...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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