A survey on rank and inertia optimization problems of the matrix-valuedfunction $A + BXB^{*}$

Yongge  Tian

doi:10.3934/naco.2015.5.289

Article Contents

2015, Volume 5, Issue 3: 289-326. Doi: 10.3934/naco.2015.5.289

This issue Previous Article Output regulation for discrete-time nonlinear stochastic optimal control problems with model-reality differences Next Article

A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$

Yongge Tian^1,

1.
CEMA, Central University of Finance and Economics, Beijing 100081

Received: March 31, 2014

Revised: June 30, 2015

Published: July 2015

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Abstract

This paper is concerned with some rank and inertia optimization problems of the Hermitian matrix-valued functions $A + BXB^{*}$ subject to restrictions. We first establish several groups of explicit formula for calculating the maximum and minimum ranks and inertias of matrix sum $A + X$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions by using some discrete and matrix decomposition methods. We then derive formulas for calculating the maximum and minimum ranks and inertias of the matrix-valued function $A + BXB^*$ subject to a Hermitian matrix $X$ that satisfies a fixed-rank and semi-definiteness restrictions, and show various properties $A + BXB^{*}$ from these ranks and inertias formulas. In particular, we give necessary and sufficient conditions for the equality $A + BXB^* = 0$ and the inequality $A + BXB^* \succ 0\, (\succeq 0, \prec 0, \, \preceq 0)$ to hold respectively for these specified Hermitian matrices $X$.

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Mathematics Subject Classification: Primary: 15A09, 15A24; Secondary: 65K10; 65K15.

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