求解四元数线性系统的一种新方法

利用矩阵半张量积以及矩阵的H-表示方法求解四元数Stein方程的循环解。首先提出了四元数矩阵的矩阵半张量积的一些新结论,进而利用这些结论将四元数Stein方程转化为具有独立变量的矩阵方程;然后利用循环矩阵的H-表示以及经典矩阵理论给出原系统循环解存在的充要条件及通解表达式;最后通过相应的数值算法验证该算法的有效性,并将该方法用于求解线性时变系统中的四元数Stein方程。

A new method for solving quaternion linear system

The circulant solution of quaternion Stein equation is solved by using semi-tensor product of matrices and H-representation method. First, some new conclusions about the semi-tensor product of quaternion matrices are presented. The quaternion Stein equation is transformed into a matrix equation with independent variables by using this conclusions. Then, the sufficient and necessary conditions for the existence of the circulant solution and the general solution expression of the original system are given by using H-representation of the circulant matrix and classical matrix theory. Finally, the effectiveness of the algorithm is verified by the corresponding numerical algorithm, and the method is applied to solve quaternion Stein equations in linear time-varying systems.