四元数Lyapunov方程的酉结构解及最佳逼近  

The unitary structure solution of quaternion Lyapunov equationand its optimal approximation

作　　者：黄敬频[1] 刘广梅 HUANG Jingpin;LIU Guangmei(College of Mathematics and Physics,Guangxi Minzu University,Nanning 530006,China)

机构地区：[1]广西民族大学数学与物理学院,南宁530006

出　　处：《重庆理工大学学报（自然科学）》2023年第8期348-354,共7页Journal of Chongqing University of Technology：Natural Science

基　　金：国家自然科学基金项目(12361078)。

摘　　要：针对四元数Lyapunov方程AX+XA*=C,在A、C为正规矩阵条件下,利用四元数矩阵的Frobenius范数酉乘积不变性和矩阵的右特征值分解,得到了该方程存在酉结构解的充分必要条件及其通解表达式。同时对预先给定的酉矩阵M,应用四元数矩阵的迹不等式和矩阵的分块方法,在该方程的酉结构解集中得到与M的最佳逼近解。最后给出求解步骤,通过数值算例和四元数矩阵的复表示运算,检验了所得理论结果的正确性及所给方法的可行性。For the quaternion Lyapunov equation AX+XA*=C,under the A,C are normal matrix condition,by using the Frobenius norm unitary product invariance of quaternion matrix and the right eigenvalue decomposition of matrix,the necessary and sufficient conditions for the existence of the unitary structure solution of the equation and its general solution expression are obtained.At the same time,for a given unitary matrix M,the best approximation solution to M is obtained in the unitary structure solution set of the equation by using the trace inequality of the quaternion matrix and the block method of the matrix.Finally,the solution steps are given,and the correctness of the theoretical results and the feasibility of the given method are verified by numerical examples and complex representation operations of the quaternion matrix.

关 键 词：四元数Lyapunov方程 右特征值分解 酉矩阵 最佳逼近 

分 类 号：O241.6[理学—计算数学]