The generalized quaternion matrix equation AXB+CX⋆D=E

In this paper, we discuss the generalized quaternion matrix equation A X B + C X ⋆ D = E , where X ⋆ is one of X , X * , the η ‐conjugate or the η ‐conjugate transpose of X with η ∈{ i , j , k }. Two new real representations of a generalized quaternion matrix are proposed. By using this method, the criteria for the existence and uniqueness of solutions to the mentioned matrix equation as well as the existence of X =± X ⋆ solutions to the generalized quaternion matrix equation A X B + C X D = E are derived in a unified way.