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Max-min problems on the ranks and inertias of the matrix expressions $$A - BXC \pm (BXC)^{\ast}$$ with applications. (English) Zbl 1223.90077
In this paper, a simultaneous decomposition for a matrix triplet $$(A,B,C ^{\ast})$$ is introduced, where $$A=\pm A^{\ast}$$ and $$(\cdot )^{\ast}$$ denotes the conjugate transpose of a matrix. Some conjectures on the maximal and minimal values of the ranks of the matrix expressions $$A - BXC \pm (BXC)^{\ast}$$ with respect to a variable matrix $$X$$ are solved. Some explicit formulas for the maximal and minimal values of the inertia of the matrix expression $$A - BXC - (BXC)^{\ast}$$ with respect to $$X$$ are given. As applications, the extremal ranks and inertias of the matrix expression $$D - CXC^{\ast}$$ subject to Hermitian solutions of a consistent matrix equation $$AXA^{\ast}=B$$, as well as the extremal ranks and inertias of the Hermitian Schur complement $$D - B^{\ast} A ^{\sim} B$$ with respect to a Hermitian generalized inverse $$A ^{\sim}$$ of $$A$$ are derived.

##### MSC:
 90C47 Minimax problems in mathematical programming 15A09 Theory of matrix inversion and generalized inverses
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##### References:
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