A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications

We give new necessary and sufficient conditions for the existence of a common solution to the pair of linear matrix equations A1XB1 = C1 and A2XB2 = C2 and derive a new representation of the general common solution to these two equations. We apply this result to determine new necessary and sufficient conditions for the existence of an Hermitian solution and a representation of the general Hermitian solution to the matrix equation AXB = C.