Mathematics Faculty Articles

Title

Some Inequalities for the Eigenvalues of the Product of Positive Semi-definite Hermitian Matrices

Document Type

Article

Publication Date

1-1-1992

Publication Title

Linear Algebra and its Applications

ISSN

0024-3795

Volume

160

Issue/No.

1

First Page

113

Last Page

118

Abstract

Let λ1(A)⩾⋯⩾λn(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i1< ⋯ <ikn. Our main results are

∑t=1kλt(GH)⩽∑t=1kλit(G)λn−it+1(H)

And

∑t=1kλit(GH)⩽∑t=1kλit(G)λn−t+1(H)

Here G and H are n by n positive semidefinite Hermitian matrices. These results extend Marshall and Olkin's inequality

∑t=1kλt(GH)⩽∑t=1kλt(G)λn−t+1(H)

We also present analogous results for singular values.

Comments

Under an Elsevier user license

DOI

10.1016/0024-3795(92)90442-D