Least squares X =± X η * solutions to split quaternion matrix equation A X A η * = B

In the paper, the split quaternion matrix equation A X A η * = B is considered, where the operator A η * is the η ‐conjugate transpose of A , where η ∈{ i , j , k }. We propose some new real representations, which well exploited the special structures of the original matrices. By using this method, we obtain the necessary and sufficient conditions for A X A η * = B to have X =± X η * solutions and derive the general expressions of solutions when it is consistent. In addition, we also derive the general expressions of the least squares X =± X η * solutions to it in case that this matrix equation is not consistent.