四元数广义Sylvester方程M自共轭混合结构解  

M self-conjugate mixed structure solutions of the quaternion generalized Sylvester equation

作　　者：徐云 黄敬频[1] XU Yun;HUANG Jingpin(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,广西南宁530006

出　　处：《河北大学学报（自然科学版）》2023年第4期337-345,共9页Journal of Hebei University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(11661011)。

摘　　要：利用矩阵的四分块形式刻画了M自共轭矩阵的特征结构,并讨论了四元数广义Sylvester方程AX-YB=C的一类M自共轭混合结构解,其中X为酉相似块对角M自共轭矩阵,Y为自共轭矩阵.根据所提结构矩阵的特点,将原方程转化为等价的无约束方程组,再利用矩阵的Moore-Penrose广义逆,获得方程组可解的充分必要条件及其通解表达式,从而得到原方程的M自共轭混合结构解.特别地,导出矩阵方程AX=C具有酉相似块对角M自共轭解的充要条件及其通解表达式.当M=0时,利用四元数矩阵对的CCD-Q分解,获得广义Sylvester方程满足‖Y~‖=min‖Y‖的约束混合结构解集.数值算例检验了所得结果的正确及可行性.In this paper,using the specific structure M self-conjugate matrix in the form of quarto-block matrix,we discuss a class of M self-conjugate mixed structure solutions of generalized Sylvester equation AX-YB=C on the quaternion field,where X is a unitary similar block diagonal M self-conjugate matrix,Y is a self-conjugate matrix.According to the characteristics of the proposed structure matrix,the original structural equation is transformed into an equivalent system of unconstrained set of equations,and the necessary and sufficient conditions for the solution of the equations and the general solution expression are obtained by using the Moore-Penrose generalized inverse of the matrix.Thus,the M self-conjugate mixed structure solution of the original equation is obtained.In particular,the necessary and sufficient conditions for matrix equation AX=C to have unitary similar block diagonal M self-conjugate solutions and the general solution expression are derived.When M=0,the constrained mixed structure solution set of generalized Sylvester equation satisfying‖Y~‖=min‖Y‖is obtained by using the CCD-Q decomposition of quaternion matrix pairs.A numerical example is given to verify the correctness and feasibility of the results.

关 键 词：四元数 广义Sylvester方程 M自共轭 混合结构解 

分 类 号：O241.6[理学—计算数学]