Solvability, Unique Solvability, and Representation of Solutions for Systems of Coupled Linear Matrix Equations

Abstract

We investigate a system of coupled linear matrix equations of the form AXB + CYD = E, CXD + AYB = F where A,B,C,D,E,F are rectangular complex matrices and X,Y are unknown complex matrices. We obtain several criterions for solvability and unique solvability of the system and tis special cases. These criterions rely on Kronecker product, vector operator, Moore Penrose inverses, and ranks. Moreover, explicit formulas of solutions are presented.