On some trace inequalities for positive definite Hermitian matrices

Houqing Zhou

doi:10.1186/1029-242x-2014-64

Abstract

Let A be a positive definite Hermitian matrix, we investigate the trace inequalities of A. By using the equivalence of the deformed matrix, according to some properties of positive definite Hermitian matrices and some elementary inequalities, we extend some previous works on the trace inequalities for positive definite Hermitian matrices, and we obtain some valuable theory. MSC:15A45, 15A57.

Highlights

A Hermitian matrix is a square matrix with complex entries that is equal to its own conjugate transpose
The problems of the trace inequality for positive definite Hermitian matrices have caught the attention of scholars, getting a lot of interesting results
In the paper, using the identical deformation of matrix, and combined with some elementary inequalities, our purpose is to derive some new results on the trace inequality for positive definite Hermitian matrices

Summary

Introduction

A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose. [ ] proved some trace inequalities for positive definite Hermitian matrices: ( ) tr(AB) ≤ tr(A B ); ( ) tr(AB) ≤ tr A + tr B ; ( ) The problems of the trace inequality for positive definite (semidefinite) Hermitian matrices have caught the attention of scholars, getting a lot of interesting results.

Results

Conclusion