A System of Matrix Equations over the Quaternion Algebra with Applications

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations A(1)X(1) = C-1, AX(1)B(1) + X2B2 = C-3, A(2)X(2) + A(3)X(3)B = C-2 and X3B3 = C-4 over the quaternion algebra H, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations AX = C, X B = C over H to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.