Abstract: Based on the real representation of the quaternion matrix, combined with the matrix H-representation and semi-tensor product of matrices, an effective method for solving the minimal norm least square Toeplitz solution of the quaternion matrix equation (A₁XB₁,…,A_kXB_k)=(C₁,…,C_k) is proposed in this paper. The necessary and sufficient condition for the existence of Toeplitz solution to the quaternion matrix equation are provided, and a general expression of solutions is also obtained. The numerical algorithm is given, and examples are given to verify the effectiveness of the method in terms of error and computation time.

Key words: quaternion matrix equation, semi-tensor product of matrices, the minimal norm least square Toeplitz solution, real representation, H-representation