Maximization and minimization of the rank and inertia of the Hermitian matrix expression A - BX - (BX)* with applications

被引：39

作者：

Tian, Yongge ^{[1
]}

机构：

[1] Cent Univ Finance & Econ, China Econ & Management Acad, Beijing 100081, Peoples R China

来源：

LINEAR ALGEBRA AND ITS APPLICATIONS | 2011年 / 434卷 / 10期

关键词：

Matrix function; Matrix equation; Rank; Inertia; Equality; Inequality; Hermitian solution; Skew-Hermitian solution; Definite solution; Re-definite solution; Lowner partial ordering; Maximization; Minimization; NONNEGATIVE DEFINITE SOLUTIONS; INVERSE PROBLEM AX; BLOCK MATRICES; OPERATOR-EQUATIONS; SCHUR COMPLEMENT; EXTREMAL RANKS; COMPLETIONS; SUBMATRICES; FORMULAS; ASTERISK;

D O I：

10.1016/j.laa.2010.12.010

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

We give in this paper a group of closed-form formulas for the maximal and minimal ranks and inertias of the linear Hermitian matrix function A - BX - (BX)* with respect to a variable matrix X. As applications, we derive the extremal ranks and inertias of the matrix X +/- X*, where X is a solution to the matrix equation AXB = C, and then give necessary and sufficient conditions for the matrix equation AXB = C to have Hermitian, definite and Re-definite solutions. In addition, we give closed-form formulas for the extremal ranks and inertias of the difference X-1 - X-2, where X-1 and X-2 are Hermitian solutions of two matrix equations A(1)X(1)A(1)* = C-1 and A(2)X(2)A(2)* = C-2. and then use the formulas to characterize relations between Hermitian solutions of the two equations. (C) 2010 Elsevier Inc. All rights reserved.

引用

页码：2109 / 2139

页数：31

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