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 \(X=\pm X^{\eta }\) solutions and \(\eta \)-(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 \(X^\star \). Finally, we present several algorithms and numerical examples to demonstrate the results of this paper.

我们设计了分裂四元数矩阵的几种实数表示，其主要目标是为分裂四元数矩阵方程组中解的存在性建立必要和充分条件。这包括没有任何约束的通解的条件，以及\(X=\pm X^{\eta }\)解和\(\eta \) -(anti-)Hermitian 解。此外，当它可解时，我们推导出通解的表达式。作为一个应用，我们研究了涉及\(X^\star \)的五个分裂四元数矩阵方程组的解。最后，我们提出了几种算法和数值例子来证明本文的结果。