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Max-min problems on the ranks and inertias of the matrix expressions A-BXC±(BXC) * with applications. (English) Zbl 1223.90077
In this paper, a simultaneous decomposition for a matrix triplet (A,B,C * ) is introduced, where A=±A * and (·) * denotes the conjugate transpose of a matrix. Some conjectures on the maximal and minimal values of the ranks of the matrix expressions A-BXC±(BXC) * 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) * with respect to X are given. As applications, the extremal ranks and inertias of the matrix expression D-CXC * subject to Hermitian solutions of a consistent matrix equation AXA * =B, as well as the extremal ranks and inertias of the Hermitian Schur complement D-B * A B with respect to a Hermitian generalized inverse A of A are derived.
MSC:
90C47Minimax problems
15A09Matrix inversion, generalized inverses
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