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Equalities and inequalities for inertias of Hermitian matrices with applications. (English) Zbl 1205.15033
This paper deals with the inertia of Hermitian matrices. Some basic formulas for inertias of \(2\times 2\) block Hermitian matrices are presented. Then several equalities and inequalities involving operations with inertias and matrices are obtained. As applications extremal values of the rank and inertia of the matrix equation \(A-BXB^*\), rank and inertia of the Hermitian solution of the matrix equation \(AX=B\), extremal values of the rank and inertia of a partial Hermitian matrix and the extremal inertias of the matrix equation \(A-B_1X_1B_{1}^*- \cdots - B_kX_kB_{k}^*\) are also presented.

MSC:
15A42 Inequalities involving eigenvalues and eigenvectors
15A24 Matrix equations and identities
15A45 Miscellaneous inequalities involving matrices
15B57 Hermitian, skew-Hermitian, and related matrices
15A03 Vector spaces, linear dependence, rank, lineability
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