Solutions with special structure to the linear matrix equation AX = B

In this article we solve the following three kinds of problems. First, some formulas for the minimal rank of the submatrices in a solution X of matrix equation AX = B and the minimal and maximal rank of X itself are derived by using the matrix rank method. From these formulas, necessary and sufficient conditions are given for X to be nonsingular or the submatrices to be zero, respectively. Second, some formulas for the minimal rank of X + X∗, X −X∗ and the corresponding expressions of submatrices of X are investigated. Combined with matrix decomposition, necessary and sufficient conditions are given for the existence of solutions to be Hermitian, local Hermitian, Skew-Hermitian and local Skew-Hermitian, respectively. Third, necessary and sufficient conditions are given for the existence of solutions to be local positive (negative) semidefinite, and for some Hermitian solution with zero submatrix, the structure form of a positive (negative) semidefinite solution are obtained using results of inertia.