Direct methods on η-Hermitian solutions of the split quaternion matrix equation (AXB,CXD)=(E,F)

This paper provides two direct methods for solving the split quaternion matrix equation (AXB,CXD)=(E,F), where X is an unknown split quaternion eta-Hermitian matrix, and A, B, C, D, E, F are known split quaternion matrices with suitable size. Our tools are the Kronecker product, Moore-Penrose generalized inverse, real representation, and complex representation of split quaternion matrices. Our main work is to find the necessary and sufficient conditions for the existence of a solution of the matrix equation mentioned above, derive the explicit solution representation, and provide four numerical algorithms and two numerical examples.