Abstract: Quaternion matrices and quaternion matrix equations play important roles in quantum mechanics. By introducing real representations of quaternion matrices, this paper studies the solution of the general quaternion matrix equation AXB-CYD=E, and gives a prectical technique of finding the solution of the quaternion matrix equation. This paper also obtains a Roth's theorem over the quaternion field.

Key words: Matrix equations, Quaternion matrices, Real representation, Solution