An Overview of Methods for Solving the System of Matrix Equations A1XB1 = C1 and A2XB2 = C2

This paper primarily investigates the solutions to the system of matrix equations $A_1 X B_1 = C_1$ and $A_2 X B_2 = C_2$. This system generalizes the classical equation $AXB = C$, as well as the system of equations $AX = B$ and $XC = D$, and finds broad applications in control theory, signal processing, networking, optimization, and other related fields. Various methods for solving this system are introduced, including the generalized inverse method, the vec-operator method, matrix decomposition techniques, Cramer's rule, and iterative algorithms. Based on these approaches, the paper discusses general solutions, symmetric solutions, Hermitian solutions, and other special types of solutions over different algebraic structures, such as number fields, the real field, the complex field, the quaternion division ring, principal ideal domains, regular rings, strongly *-reducible rings, and operators on Banach spaces. In addition, matrix systems related to the system $A_1 X B_1 = C_1$ and $A_2 X B_2 = C_2$ are also explored.