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Zbl 1211.15022
Tian, Yongge
Maximization and minimization of the rank and inertia of the Hermitian matrix expression $A-BX-(BX)^{*}$ with applications. (English)
[J] Linear Algebra Appl. 434, No. 10, 2109-2139 (2011). ISSN 0024-3795

The author gives 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, he derives the extremal ranks and inertias of the matrices $X\pm X^{*}$, where $X$ is a solution to the matrix equation $AXB=C$, and then he gives necessary and sufficient conditions for the matrix equation $AXB=C$ to have Hermitian, definite and re-definite solutions. In addition, he gives 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 the two matrix equations $A_1 X_1 A_1^* = C_1$ and $A_2 X_2 A_2^* = C_2$, and then uses the formulas to characterize relations between Hermitian solutions of the two equations.
[A. Arvanitoyeorgos (Patras)]

MSC 2000:: *15A24 Matrix equations
15A03 Vector spaces
15A42 Inequalities involving eigenvalues and eigenvectors
15B57

Keywords: matrix function; matrix equation; inertia; Hermitian matrix; Löwner partial ordering; maximization; minimization; ranks; Hermitian solutions

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