四元数矩阵的直积分解及最佳逼近  

Kronecker Product Decomposition of Quaternion Matrix and Its Optimal Approximation

作　　者：黄敬频[1] 白瑞 徐云 赵耿威 HUANG Jingpin;BAI Rui;XU Yun;ZHAO Gengwei(School of Mathematics and Physics, Guangxi University for Nationalities, 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：《西南师范大学学报（自然科学版）》2022年第2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-chxps202071).

摘　　要：讨论了直积意义下四元数矩阵的分解问题,即对于给定的四元数矩阵A,讨论是否存在两个四元数矩阵X,Y,满足A=X■Y,同时给出A的二次方根的存在条件及计算方法.首先利用A的分块矩阵及其拉直矩阵的秩,获得A具有Kronecker积分解的充要条件及分解方法.当此类分解不存在时,利用拉直矩阵的奇异值分解得到相应的最佳逼近分解.然后应用直积的定义导出了X■X=A成立的充要条件及二次方根X的计算公式.最后通过两个数值算例,检验了所给方法的有效性及可行性.The decomposition problem of a quaternion matrix has been discussed in Kronecker product sense,namely for a given quaternion matrix A,whether two quaternion matrices X,Y exists so as to A=X■Y,and the existence conditions of square root of A and its calculation method have been found out.In the whole process,it is mainly used partitioning of A and rank of a vectorized matrix,the necessary and sufficient conditions exists Kronecker product decomposition of A and its decomposition method have been obtained.When the above decomposition does not exist,optimal approximation is given by singular value decomposition of a vectorized matrix.Meanwhile,the conditions for equality X■X=A and the formula for calculating square root X has been deduced by the definition of Kronecker product.Finally,two simulation examples have been given to illustrate the validity and feasibility of the method.

关 键 词：四元数矩阵 Kronecker积分解 秩 最佳逼近 二次方根 

分 类 号：O151.21[理学—数学]