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Maximization and minimization of the rank and inertia of the Hermitian matrix expression $A-BX-{\left(BX\right)}^{*}$ with applications. (English) Zbl 1211.15022
The author gives a group of closed-form formulas for the maximal and minimal ranks and inertias of the linear Hermitian matrix function $A-BX-{\left(BX\right)}^{*}$ with respect to a variable matrix $X$. As applications, he derives the extremal ranks and inertias of the matrices $X±{X}^{*}$, 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.
##### MSC:
 15A24 Matrix equations and identities 15A03 Vector spaces, linear dependence, rank 15A42 Inequalities involving eigenvalues and eigenvectors 15B57 Hermitian, skew-Hermitian, and related matrices