A simple method for solving matrix equations $ AXB = D $ and $ GXH = C $

A simple method to solve the common solution to the pair of linear matrix equations $ AXB = D $ and $ GXH = C $ is introduced. Some necessary and sufficient conditions for the existence of a common solution, and two expressions for the general common solution of the equation pair are provided by the proposed method. Subsequently, the results are applied to determine the solution of the matrix equation $ AXB+GYH = D $ and the Hermitian solution of the matrix equation $ AXB = D. $