On the split quaternion matrix equation $$AX=B$$

Abstract

In the paper, we consider the split quaternion matrix equation $$AX=B$$. We design several real representations of split quaternion matrix to transform the above split quaternion matrix equation into some real matrix equations. By using this method, we give some necessary and sufficient conditions for $$AX=B$$ to have a X or $$X=\pm X^{\star }$$ solution and derive the expressions of solutions when equation is solvable, where $$X^{\star } \in \{X^*, X^\eta , X^{\eta *}\}$$, $$X^*$$ is the conjugate transpose of X, for $$\eta \in \{i, j, k\}$$, $$ X^{\eta }, X^{\eta *}$$ are $$\eta $$-conjugate, $$\eta $$-Hermitian of X, respectively. We also present the solvability conditions and expression of the unique solution X or $$X=\pm X^{\star }$$.