A Classical System of Matrix Equations Over the Split Quaternion Algebra

We design several real representations of split quaternion matrices with the primary objective of establishing both necessary and sufficient conditions for the existence of solutions within a system of split quaternion matrix equations. This includes conditions for the general solution without any constraints, as well as \(X=\pm X^{\eta }\) $X=\pm X^{\eta }$ solutions and \(\eta \) $\eta$ -(anti-)Hermitian solutions. Furthermore, we derive the expressions for the general solutions when it is solvable. As an application, we investigate the solutions to a system of five split quaternion matrix equations involving \(X^\star \) $X^{\star }$ . Finally, we present several algorithms and numerical examples to demonstrate the results of this paper.