59 can serve as right hand singular vectors for a but

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Unformatted text preview: 1 ≥ (x∗ Ax/x∗ x) ≥ λn for all x = 0. The term x∗ Ax/x∗ x is referred to as a Rayleigh quotient in honor of the famous English physicist John William Strutt (1842–1919) who became Baron Rayleigh in 1873. It’s only natural to wonder if the intermediate eigenvalues of a hermitian matrix have representations similar to those for the extreme eigenvalues as described in (7.5.4). Ernst Fischer (1875–1954) gave the answer for matrices in 1905, and Richard Courant (1888–1972) provided extensions for infinite-dimensional operators in 1920. D E Courant–Fischer Theorem The eigenvalues λ1 ≥ λ2 ≥ · · · ≥ λn of a hermitian matrix An×n are λi = max dim V =i min x∗ Ax and x∈V x 2 =1 λi = min dim V =n−i+1 T H max x∗ Ax. x∈V x 2 =1 (7.5.5) When i = 1 in the min-max formula and when i = n in the maxmin formula, V = C n , so these cases reduce to the equations in (7.5.4). Alternate max-min and min-max formulas are given in Exercise 7.5.12. IG R Proof. Only the min-max characterization is proven—the max-min proof is analogous (Exercise 7.5.11). As shown in Example 7.5.1, a change of coordinates y = U∗ x with a unitary U such that U∗ AU = D = diag (λ1 , λ2 , . . . , λn ) has the effect of replacing A by D, so we need only establish that Y P λi = min dim V =n−i+1 O C max y∗ Dy. y∈V y 2 =1 For a subspace V of dimension n − i + 1, let SV = {y ∈ V , y SV = {y ∈ V ∩ F , y 2 = 1}, where 2 = 1}, and let F = span {e1 , e2 , . . . , ei } . Note that V ∩ F = 0, for otherwise dim(V + F ) = dim V + dim F = n + 1, which is impossible. In other words, SV contains those vectors of SV of the i 2 form y = (y1 , . . . , yi , 0, . . . , 0)T with j =1 |yj | = 1. So for each subspace V with dim V = n − i + 1, i y∗ Dy = i λj |yj |2 ≥ λi j =1 |yj |2 = λi for all y ∈ SV . j =1 Since SV ⊆ SV , it follows that maxSV y∗ Dy ≥ maxSV y∗ Dy ≥ λi , and hence min max y∗ Dy ≥ λi . V Copyright c 2000 SIAM SV Buy online from SIAM...
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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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