Several matrix trace inequalities on Hermitian and skew-Hermitian matrices

Xiangyu Gao,Xian Zhang,Julong Tan,Guoqiang Wang

doi:10.1186/1029-242x-2014-358

Abstract

In this paper, we present several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing interior-point methods (IPMs) for semidefinite optimization (SDO).

Highlights

1 Introduction semidefinite optimization (SDO) is the generalization of linear optimization (LO), which is convex optimization over the intersection of an affine set and the cone of positive semidefinite matrices
Several interior-point methods (IPMs) designed for LO have been successfully extended to SDO [, ]
In [ ], Yang proved the arithmetic mean-geometric mean inequality for positive definite matrices, which was an open question proposed by Bellman in [ ]; Neudecke used a different method in [ ] to show a slightly relaxed version of Yang’s result in [ ]; In [ ], Coope considered alternative proofs of some simple matrix trace inequalities in [ – ] and further studied properties of products of Hermitian and positivedefinite matrices; In [ ], Yang gave a new proof of the result obtained by Yang in [ ] and extended it to a generalized positive semidefinite matrix

Summary

Introduction

SDO is the generalization of linear optimization (LO), which is convex optimization over the intersection of an affine set and the cone of positive semidefinite matrices. Some matrix trace inequalities are developed and applied in the analysis of IPMs for SDO (see [ – ]). Based on the work in [ – ], Chang established a matrix trace inequality for products of Hermitian matrices in [ ], which partly answers a conjecture proposed by Bellman in [ ]. We will provide several matrix trace inequalities on Hermitian and skew-Hermitian matrices, which play an important role in designing and analyzing IPMs for SDO.

Results

Conclusion