Semi-tensor Product Method of η-Hermitian Solution of Split Quaternion Matrix Equation X+AXB=C

In order to study the structure of the split quaternion matrix equation X+AXB=C, a universal method is proposed for solving the special solution of the split quaternion matrix equations, and a real vector representation of the split quaternions is proposed. The η-Hermitian solution of the split quaternion matrix equation X+AXB=C is studied by combining the matrix semi-tensor product. Firstly, the matrix semi-tensor product and the real vector representations of the split quaternion matrices A, B and C are utilized to derive the real vector representations of the split quaternion matrices A+B, cA, AB and ABC. And the split quaternion matrix equation is transformed into an equivalent real matrix equation form. Then, utilizing the special structure of the split quaternion η-Hermitian matrix and the relevant theory of matrix semi-tensor product, independent elements are extracted to simplify the computation. Finally,leveraging the necessary and sufficient conditions for the existence of a solution to the real matrix equation Ax=b, the necessary and sufficient conditions and general solution for the existence of η-Hermitian solutions for the split quaternion matrix equation are obtained. The validity of the method has been verified with the aid of numerical examples. The study of η-Hermitian solutions for the split quaternion matrix equation X+AXB=C in this article is a supplement to matrix theory and provides new insights for solving special solutions of matrix equations.