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

Authors

Fuzhen Zhang, Nova Southeastern UniversityFollow
Bo-Ying Wang, Beijing Normal University

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)⩾⋯⩾λ_n(A) denote the eigenvalues of a Hermitian n by n matrix A, and let 1⩽i₁< ⋯ <i_k⩽n. 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

NSUWorks Citation

Zhang, Fuzhen and Wang, Bo-Ying, "Some Inequalities for the Eigenvalues of the Product of Positive Semi-definite Hermitian Matrices" (1992). Mathematics Faculty Articles. 111.
https://nsuworks.nova.edu/math_facarticles/111

DOI

10.1016/0024-3795(92)90442-D