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Solvability of systems of linear matrix equations subject to a matrix inequality. (English) Zbl 1357.15009

The authors study the solvability conditions and the explicit expressions of the Hermitian solutions to the system of matrix equations \(AX=B\), \(XC=D\) subject to a matrix inequality \(MXM^* \geq N\) and the Hermitian nonnegative definite solutions to the system of matrix equations \(AX=B\), \(XC=D\) subject to a matrix inequality \(MXM^* \geq N\geq 0\). They make full use of the generalized inverse and the rank of matrices. As applications, some special cases of the above systems of matrix equations are considered. In addition, the maximal ranks and inertias of the Hermitian solutions are presented.

MSC:

15A24 Matrix equations and identities
15B57 Hermitian, skew-Hermitian, and related matrices
15B48 Positive matrices and their generalizations; cones of matrices
15A03 Vector spaces, linear dependence, rank, lineability
15A45 Miscellaneous inequalities involving matrices
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