Solvability of systems of linear matrix equations subject to a matrix inequality

In this paper, 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* >= N and the Hermitian nonnegative definite solutions to the system ofmatrix equations AX = B, XC = D subject to a matrix inequality MXM* >= N >= 0 are, respectively, put forward, by making 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, respectively, presented.