矩阵半张量积在求解四元数线性系统中的应用

Application of semi-tensor product of matrices in solving quaternion linear system

近年来,矩阵半张量积被广泛应用于布尔网络、混合值逻辑网络、电力系统非线性鲁棒稳定控制代数问题等的分析与控制.该文提出了它在四元数线性系统中的一种新的应用.利用矩阵半张量积、四元数矩阵的实向量表示和四元数三对角Hermitian(反Hermitian)矩阵的特殊结构,得到了四元数矩阵方程(AXB,CXD)=(E,F)的最小二乘三对角Hermitian(反Hermitian)解的表达式.给出了四元数矩阵方程相容的充要条件以及在相容条件下的通解表达式.还给出了数值算法,并通过实验验证了该方法的有效性.

In recent years,the semi-tensor product of matrices is widely applied to the analysis and control of Boolean network,mix-valued logical network,power system nonlinear robust stability control algebra problem and so on.In this paper,a new kind of applications in quaternion linear system is proposed.By using semi-tensor product,a real vector representation of quaternion,and the special structure of quaternion tridiagonal Hermitian(Anti-Hermitian)matrix,the expression of the least squares tridiagonal Hermitian(Anti-Hermitian)solution of(AXB,CXD)=(E,F)is obtained.As the special case,the equivalent condition for the compatibility of the quaternion matrix equation and the expression of general solutions under the compatibility conditions are given.In addition,the numerical algorithm and experiment are given to verify the effectiveness of this application.