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Maximization and minimization of the rank and inertia of the Hermitian matrix expression $$A-BX-(BX)^{*}$$ 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 $$A-BX-(BX)^{*}$$ 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.

##### MSC:
 15A24 Matrix equations and identities 15A03 Vector spaces, linear dependence, rank, lineability 15A42 Inequalities involving eigenvalues and eigenvectors 15B57 Hermitian, skew-Hermitian, and related matrices
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##### References:
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