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
收藏
页码：2109 / 2139
页数：31
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