四元数矩阵方程X^(2)+BX+XB*+Q=0的Hermite正定解  

Hermite positive definite solution of the quaternionmatrix equation X^(2)+BX+XB*+Q=0

作　　者：姚祎雯 黄敬频[1] YAO Yi-wen;HUANG Jing-pin(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：《陕西科技大学学报》2023年第5期195-202,共8页Journal of Shaanxi University of Science & Technology

基　　金：国家自然科学基金项目(12361078)。

摘　　要：讨论四元数体上二次矩阵方程X^(2)+BX+XB*+Q=0(Q>0)存在Hermite正定解的必要和充分条件及其迭代求解方法.主要针对系数矩阵的特点,通过引入适当的参数建立矩阵不等式,利用凸集上的不动点理论,证明了该方程存在Hermite正定解的一些必要和充分条件.在此基础上,对不同的条件和解存在区间构建出三种收敛的迭代格式,根据每种迭代特性给出了初始矩阵的选取方法,并运用四元数矩阵的复化算子建立Matlab环境下求解算法.与此同时对方程的解进行了扰动分析,获得2个扰动误差界.三个数值算例检验了所给方法的有效及可行性.This paper discuss necessary and sufficient conditions for the existence of Hermite positive definite solutions of the quadratic matrix equation X^(2)+BX+XB*+Q=0 on the quaternion field and its iterative solution method.Some necessary and sufficient conditions for the existence of Hermite positive definite solutions of this equation are proved mainly for the characteristics of coefficient matrices by introducing appropriate parameters to establish matrix inequalities and by using the theory of fixed points on convex sets.On this basis,three convergent iteration formats are constructed for different conditions and solution existence intervals,and the selection of the initial matrix is given according to each iteration property.The algorithm is solved in the Matlab environment by using the complexization operator of the quaternion matrix.At the same time,a perturbation analysis is carried out on the solution of the equation,and two perturbation error bounds are obtained.Three numerical examples are used to test the effectiveness and feasibility of the given method.

关 键 词：四元数 二次矩阵方程 HERMITE正定解 矩阵不等式 迭代 扰动分析 

分 类 号：O241.6[理学—计算数学]