Consistency of Split Quaternion Matrix Equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^\star B=CY+D$$

Abstract

In this paper, the split quaternion matrix equations $$AX^{\star }-XB=CY+D$$ and $$X-AX^{\star }B=CY+D$$ are considered, where $$X^\star $$ is X or the $$\phi $$ -conjugate of X, $$\phi \in \{i, j, k\}$$ . We design some new real representations of a split quaternion matrix, which enable us to convert $$\phi $$ -conjugate matrix equations into some real matrix equations. By using this method, we show that the original split quaternion matrix equation is consistent if and only if its corresponding real matrix equation is consistent. Moreover, we formulate the split quaternion solutions by the real solutions of its corresponding real matrix equation. As applications, we derive the necessary and sufficient conditions for the existence of solutions to the above two split quaternion matrix equations and their special cases $$AX^{\star }-XB=C$$ and $$X-AX^{\star }B=C$$ . We also obtain some conditions for the mentioned four split matrix equations to be uniquely solvable.