The general solution to a system of real split quaternion matrix equations

Abstract

In this paper, we present a direct methodology for solving a novel system of split quaternion matrix equations. Leveraging the Moore-Penrose generalized inverse, the Kronecker product, the $\mathrm{vec}$ operator, and the real representation of split quaternion matrices, we offer a comprehensive toolkit. The primary aim of this paper is to establish the solvability conditions of the system over the split quaternions and provide a general solution expression when it is consistent. We also give an algorithm to find the approximate solution to this system when it is inconsistent. Finally, we offer a numerical example to showcase the efficacy of our approach.