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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±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:
15A24Matrix equations and identities
15A03Vector spaces, linear dependence, rank
15A42Inequalities involving eigenvalues and eigenvectors
15B57Hermitian, skew-Hermitian, and related matrices
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