四元数体上方程AXB=C的D自共轭解及最小二乘问题

The D-Self-conjugate Solution and Least Squares Pro b lem of AXB =C over Quaternion Field

矩阵方程AXB=C具有广泛的实际应用背景,笔者在四元数体上讨论它的D自共轭解及其最小二乘问题.首先,对于给定的四元数正定矩阵D,借助四元数向量内积,给出了D自共轭矩阵的定义.然后,利用四元数矩阵对分解定理,得到了方程AXB=C具有D自共轭解的充要条件及其解的表达式.最后,利用四元数矩阵对的广义奇异值分解,获得该方程的最小二乘D自共轭解,并通过数值算例显示该文的具体算法.所得结果推广了复域上的相关结论.

The matrix equation AXB = C has wide practical background, this paper discusses its D - Self-conjugate solution and least squares problem over quaternion field. Firstly, by using inner product of quaternion vectors along with a positive definite matrix D, the definition of D -self-conjugate matrix is given. Then, use decomposition theorem on the quaternion matrix pairs, the necessary and sufficient conditions for the existence of D -Self-conjugate matrix solutions for AXB =C are obtained together with the general forms of such solutions. In addition, the least square D-Self-conjugate solutions for AXB =C has been derived by generalized singular value decomposition on the quaternion matrix pairs, and a numerical example demonstrates the feasibility of the proposed method. The results promoted some known conclusions of complex fields.