四元数矩阵方程AXB+CXD=E的M自共轭解  

On M Self-Conjugate Solution of Quaternion Equation AXB+CXD=E

作　　者：蓝家新 黄敬频 王敏 毛利影 LAN Jia-xin;HUANG Jing-pin;WANG Min;MAO Li-ying(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：《西南师范大学学报（自然科学版）》2019年第8期1-6,共6页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：国家自然科学基金项目(11661011);广西民族大学研究生创新项目(gxun-chxzs2017142,gxun-chxps201813)

摘　　要：把实数域上的M对称矩阵的概念推广到四元数体上,形成M自共轭矩阵,然后在四元数体上讨论矩阵方程AXB+CXD=E的M自共轭解及其最佳逼近问题.利用四元数矩阵的实分解和复分解,以及M自共轭矩阵的特征结构,借助Kronecker积把约束四元数矩阵方程转化为实数域上的无约束方程,克服了四元数乘法非交换运算的困难,并得到该方程具有M自共轭解的充要条件及其通解表达式.同时在解集非空的条件下,运用矩阵的分块技术及矩阵的拉直算子,获得与预先给定的四元数矩阵有极小Frobenius范数的最佳逼近解.由于M自共轭矩阵是四元数自共轭矩阵的推广,因此所得结果拓展了该方程的结构解类型.This paper aims at extending the concept of M symmetric matrix on real number field to the formation of M self-conjugate matrix on quaternion field and discussing M self-conjugate matrix solution of quaternion equation AXB+CXD =E and its optimal approximation. With the complex and real representations of a quaternion matrix, the Kronecker product of matrices and the specific structure of a M self-conjugate matrix, the quaternion equation with constraints can be converted to an unconstrained equation and to overcome the difficulty of non-commutative operation of quaternion multiplication. Then the necessary and sufficient conditions for the existence of the quaternion matrix equation AXB+CXD=E with M self-conjugate matrix and its general solution expression have been obtained. Meanwhile under the condition of the solution set of the M self-conjugate is not empty, by applying block matrix technology and matrix vec operator, and the expression of the optimal approximation solution to the given quaternion matrix is derived. Since M self-conjugate matrix is a generalization of self-conjugate quaternion matrix, the obtained results extend the type of structural solutions of this equation. Finally, we provide numerical algorithms and numerical examples to exemplify the results.

关 键 词：四元数体 矩阵方程 M自共轭矩阵 KRONECKER积 最佳逼近 

分 类 号：O151.21[理学—数学]