Ec securehostcomsiamot71html buy from amazoncom 556

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: = 1, we have x∗ Dx ≤ λi , and βi = min dim V =n−i+1 ˜ ˜ max x∗ Bx ≤ max x∗ Bx = max x∈V x 2 =1 x∈T x 2 =1 x∈T x 2 =1 ˜ x∗ Dx + x∗ Ex ˜ ˜ ≤ max x∗ Dx + max x∗ Ex ≤ λi + max x∗ Ex = λi + n x∈T x 2 =1 Copyright c 2000 SIAM x∈T x 2 =1 x∈C x 2 =1 1. Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 552 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 Note: Because E often represents an error, only E (or an estimate thereof) is known. But for every matrix norm, | j | ≤ E for each j (Example 7.1.4, p. 497). Since the j ’s are real, − E ≤ j ≤ E , so (7.5.6) guarantees that It is illegal to print, duplicate, or distribute this material Please report violations to meyer@ncsu.edu λi − E ≤ βi ≤ λi + E . (7.5.7) In other words, • the eigenvalues of a hermitian matrix A are perfectly conditioned because a hermitian perturbation E changes no eigenvalue of A by more than E . D E It’s interesting to compare (7.5.7) with the Bauer–Fike bound of Example 7.3.2 (p. 528). When A is hermitian, (7.3.10) reduces to minλi ∈σ(A) |β − λi | ≤ E because P can be made unitary, so, for induced matrix norms, κ(P) = 1. The two results differ in that Bauer–Fike does not assume E and B are hermitian. T H Example 7.5.3 Interlaced Eigenvalues. For a hermitian matrix A ∈ C n×n with eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn , and for c ∈ C n×1 , let B be the bordered matrix B= A c∗ c α IG R with eigenvalues n+1×n+1 β1 ≥ β2 ≥ · · · ≥ βn ≥ βn+1 . Problem: Explain why the eigenvalues of A interlace with those of B in that Y P β1 ≥ λ1 ≥ β2 ≥ λ2 ≥ · · · ≥ βn ≥ λn ≥ βn+1 . (7.5.8) Solution: To see that βi ≥ λi ≥ βi+1 for 1 ≤ i ≤ n, let U be a unitary matrix such that UT AU = D = diag (λ1 , λ2 , . . . , λn ) . Since V = U 0 is 01 also unitary, the eigenvalues of B agree with those of O C ˜ B = V...
View Full Document

This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

Ask a homework question - tutors are online