A System of Four Matrix Equations over von Neumann Regular Rings and Its Applications

We consider the system of four linear matrix equations A 1 X = C 1, XB 2 = C 2, A 3 XB 3 = C 3 and A 4 XB 4 = C 4 over R, an arbitrary von Neumann regular ring with identity. A necessary and sufficient condition for the existence and the expression of the general solution to the system are derived. As applications, necessary and sufficient conditions are given for the system of matrix equations A 1 X = C 1 and A 3 X = C 3 to have a bisymmetric solution, the system of matrix equations A 1 X = C 1 and A 3 XB 3 = C 3 to have a perselfconjugate solution over with an involution and char 2, respectively. The representations of such solutions are also presented. Moreover, some auxiliary results on other systems over R are obtained. The previous known results on some systems of matrix equations are special cases of the new results.