On Relationships between a Linear Matrix Equation and Its Four Reduced Equations

Given the linear matrix equation AXB=C, we partition it into the form A1X11B1+A1X12B2+A2X21B1+A2X22B2=C, and then pre- and post-multiply both sides of the equation by the four orthogonal projectors generated from the coefficient matrices A1, A1, B1, and B2 to obtain four reduced linear matrix equations. In this situation, each of the four reduced equations involves just one of the four unknown submatrices X11, X12, X21, and X22, respectively. In this paper, we study the relationships between the general solution of AXB=C and the general solutions of the four reduced equations using some highly selective matrix analysis tools in relation to ranks, ranges, and generalized inverses of matrices.