具有箭形矩阵约束的四元数Sylvester方程求解  

The Solutions of Quaternion Sylvester Equation with Arrowhead Matrix Constraints

作　　者：黄敬频 蓝家新 毛利影 王敏 HUANG Jing-pin;LAN Jia-xin;MAO Li-ying;WANG Min(College of Science,Guangxi University for Nationalities,Nanning 530006,China)

机构地区：[1]广西民族大学理学院

出　　处：《数学的实践与认识》2018年第16期264-271,共8页Mathematics in Practice and Theory

基　　金：国家自然科学基金(11661011);广西民族大学研究生创新项目(gxun-chxzs2017142,gxun-chxzs2018035)

摘　　要：箭形矩阵是一类结构简单应用广泛的特殊矩阵,在四元数体上讨论Sylvester方程的箭形矩阵解及其最佳逼近问题.利用四元数矩阵的实分解和箭形矩阵的特征结构,借助Kronecker积把约束四元数矩阵方程转化为实域上无约束方程,从而得到四元数Sylvester方程AX-XB=C具有一般箭形解和自共轭箭形解的充要条件及其通解表达式.同时在相应的解集合中,获得与预先给定的四元数箭形矩阵有极小Frobenius范数的最佳逼近解.Arrowhead matrix with simple structure, is a special matrix and applied in many field. In this paper, we discuss the problems of solving quaternion Sylvester equation with arrowhead matrix constraints and its optimal approximation. By using the real represen- tation of a quaternion matrix and the specific structure of an arrowhead matrix, Sylvester equation with arrowhead matrix constraints can be converted to an unconstrained equation by Kronecker product. Then the necessary and sufficient condition for the existence of an arrowhead matrix solution, a self-conjugate arrowhead matrix solution and their general so- lutions of the quaternion Sylvester equation AX - XB = C are obtained. Meanwhile, in the corresponding solution set, the optimal approximation solution which has minimal Frobenius norm for given quaternion arrowhead matrix is derived.

关 键 词：四元数体 SYLVESTER方程 箭形矩阵 自共轭 最佳逼近 

分 类 号：O241.6[理学—计算数学]