On the solutions of two systems of quaternion matrix equations

We study the system of quaternion generalized Sylvester matrix equations , and . We establish necessary and sufficient conditions for the system to be solvable, and when such conditions are met, we present the general solution to the system. We also investigate the number of solutions a solvable system could have. Our techniques entail the investigation of the quaternion matrix system and in X and Y with the constraint that X and Y have a common submatrix. For such systems, we establish a characterization of solvable systems and provide their general solution. We also study in detail the number of solutions a solvable system could have and present necessary and sufficient conditions for a solvable system to have a unique solution. We finally illustrate our techniques in solving the generalized Sylvester matrix equations by providing an example.