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Unformatted text preview: ∗ BV = D y∗ y α , y = U∗ c. where For x ∈ F = span {e1 , e2 , . . . , ei } ⊂ C n+1×1 with x 2 = 1, n x = (x1 , . . . , xi , 0, . . . , 0)T and ˜ x∗ Bx = n λj |xj |2 ≥ λi j =1 |xj |2 = λi , j =1 ˜ so applying the max-min part of the Courant–Fisher theorem to B yields βi = max dim V =i Copyright c 2000 SIAM ˜ ˜ min x∗ Bx ≥ min x∗ Bx ≥ λi . x∈V x 2 =1 x∈F x 2 =1 Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 7.5 Normal Matrices http://www.amazon.com/exec/obidos/ASIN/0898714540 For x ∈ T = span{ei−1 , ei , . . . , en } ⊂ C n+1×1 with x 2 553 = 1, n n ˜ x = (0, . . . , 0, xi−1 , . . . , xn , 0)T and x∗ Bx = λj |xj |2 ≤ λi−1 j =i−1 |xj |2 = λi−1 , j =i so the min-max part of the Courant–Fisher theorem produces It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] βi = min dim V =n−i+2 ˜ ˜ max x∗ Bx ≤ max x∗ Bx ≤ λi−1 . x∈V x 2 =1 x∈F x 2 =1 D E Note: If A is any n × n principal submatrix of B, then (7.5.8) still holds because each principal submatrix can be brought to the upper-left-hand corner by a similarity transformation PT BP, where P is a permutation matrix. In other words, • the eigenvalues of an n + 1 × n + 1 hermitian matrix are interlaced with the eigenvalues of each of its n × n principal submatrices. m×n m×n T H ∗ For A ∈ C (or ), the products A A and AA∗ (or AT A and AA ) are hermitian (or real symmetric), so they are diagonalizable by a unitary (or orthogonal) similarity transformation, and their eigenvalues are necessarily real. But in addition to being real, the eigenvalues of these matrices are always nonnegative. For example, if (λ, x) is an eigenpair of A∗ A, then 2 2 λ = x∗ A∗ Ax/x∗ x = Ax 2 / x 2 ≥ 0, and similarly for the other products. In fact, these λ ’s are the squares of the singular values for A developed in §5.12 (p. 411) because if Dr×r 0 A=U V∗ 0 0 m×n IG R T Y P O C is a singular value decomposition, where D = diag (σ1 ,...
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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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