四元数矩阵方程(A_(1)XB_(1),…,A_(k)XB_(k))=(C_(1),…,C_(k))的极小范数最小二乘Toeplitz解

Minimal norm least square Toeplitz solution of quaternion matrix equation(A_(1)XB_(1),…,A_(k)XB_(k))=(C_(1),…,C_(k))

基于四元数矩阵实表示,结合矩阵H-表示和矩阵半张量积提出一种求解四元数矩阵方程(A_(1)XB_(1),…,A_(k)XB_(k))=(C_(1),…,C_(k))的极小范数最小二乘Toeplitz解的有效方法,给出该四元数矩阵方程存在Toeplitz解的充要条件及通解表达式.给出数值算法并通过算例分别从误差与计算时间两个方面验证该方法的有效性.

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(A_(1)XB_(1),…,A_(k)XB_(k))=(C_(1),…,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.