四元数矩阵方程AX=B的共轭次辛解及其逼近  

On Conjugate Sub-symplectic Matrix Solutions of the Quaternion Equation AX=B and Its Optimal Approximation

作　　者：蓝家新 黄敬频[2] 黄丹 吴发乾 LAN Jiaxin;HUANG Jingpin;HUANG Dan;WU Faqian(School of Mathematics and Statistics,Baise University,Baise Guangxi 533000,China;College of Mathematics and Physics,Guangxi University for Nationalities,Nanning 530006,China)

机构地区：[1]百色学院数学与统计学院,广西百色533000 [2]广西民族大学数学与物理学院,南宁530006

出　　处：《西南师范大学学报（自然科学版）》2021年第11期8-14,共7页Journal of Southwest China Normal University(Natural Science Edition)

基　　金：广西高校中青年教师科研基础能力提升项目(2020KY19014).

摘　　要：研究了四元数矩阵方程AX=B的共轭次辛解及其逼近问题.利用共轭转置矩阵与共轭次转置矩阵的联系、四元数矩阵的实分解及矩阵Kronecker积,将约束方程转化为实数域上无约束方程组,从而得到四元数矩阵方程AX=B具有共轭次辛矩阵解的充要条件及其通解表达式.同时在共轭次辛解集中找到与给定共轭次辛矩阵有极小Frobenius范数的最佳逼近解.最后给出2个数值算例表明该算法的可行性.In this paper,we discuss the conjugate sub-symplectic matrix solutions of the quaternion equation AX=B and its optimal approximation.With the relationship between conjugate transpose matrix and conjugate secondary transpose matrix,the real decomposition of a quaternion matrix,and the Kronecker product of matrices,the quaternion equation with constraints can be converted to unconstrained equations.Then the necessary and sufficient condition for the existence of the quaternion matrix equation AX=B with conjugate sub-symplectic matrix and its general solution expression is obtained.Meanwhile under the condition of the solution set of the conjugate sub-symplectic matrix is not empty,and the expression of the optimal approximation solution to the given quaternion matrix is derived.Finally,numerical examples are presented to show the effectiveness of our algorithm.

关 键 词：四元数体 矩阵方程 共轭次转置 共轭次辛矩阵 最佳逼近 

分 类 号：O151.21[理学—数学]