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Article
Some Inequalities for the Eigenvalues of the Product of Positive Semi-definite Hermitian Matrices
Linear Algebra and its Applications
  • Fuzhen Zhang, Nova Southeastern University
  • Bo-Ying Wang, Beijing Normal University
Document Type
Article
Publication Date
1-1-1992
Disciplines
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

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DOI
10.1016/0024-3795(92)90442-D
Citation Information
Fuzhen Zhang and Bo-Ying Wang. "Some Inequalities for the Eigenvalues of the Product of Positive Semi-definite Hermitian Matrices" Linear Algebra and its Applications Vol. 160 Iss. 1 (1992) p. 113 - 118 ISSN: 0024-3795
Available at: http://works.bepress.com/fuzhen-zhang/2/