The (anti-) η -Hermitian solution to a novel system of matrix equations over the split quaternion algebra1

In comparison to quaternions, split quaternions exhibit a more intricate algebraic structure, characterized by the presence of nontrivial zero factors. Furthermore, in various fields such as geometry and electromagnetism, split quaternions serve as more convenient research tools than quaternions. This paper aims to establish the necessary and sufficient conditions for the existence of (anti-) η -Hermitian solutions to a novel system of matrix equations formulated over split quaternions. We also provide the general expression of such solutions when the system is solvable, along with the least squares (anti-) η -Hermitian solution in the case of inconsistency. To illustrate the key findings of this paper, numerical examples are presented.