关于非线性矩阵方程X^s-A₁*X^-t₁A₁-A₂*X^-t₂A₂=Q的若干结果  

Some Results on the Nonlinear Matrix Equation Xs-A1*X-t1A1-A2*X-t2A2=Q

作　　者：裴伟娟 

机构地区：[1]河南开封科技传媒学院,河南 开封

出　　处：《理论数学》2023年第12期3798-3802,共5页Pure Mathematics

摘　　要：本文主要研究非线性矩阵方程Xs− A1∗X−t1A1−A2∗X−t2A2=Q的正定解,其中A1、A2为n×n复矩阵,s、t1、t2为正整数,Q为n×n正定矩阵。文中将矩阵方程等价变形后,基于线性方程组在系数矩阵非列满秩时有非零解和矩阵特征值、特征向量的定义,研究了该矩阵方程正定解的最大和最小特征值的性质。通过正定矩阵的Cholesky分解给出了该非线性矩阵方程存在Hermitian正定解的新的充分必要条件。In this paper, we mainly investigate the Hermitian positive definite solution of the nonlinear matrix equation Xs− A1∗X−t1A1−A2∗X−t2A2=Q, where A1,A2 are n×n complex matrices, s、t1、t2 are positive integers and Q is an n×n positive definite matrix. Firstly, by equivalent deformation of the matrix equation, with the help of the theory that linear equations have non-zero solutions when the coefficient matrix is not full columnrank, and the definition of matrix eigenvalue and ei-genvector, the property of the maximum and minimum eigenvalues of positive definite solutionsis studied. Next, by Cholesky decomposition of positive definite matrix, a new sufficient and necessary condition for the equation to have a positive definite solution is proposed.

关 键 词：非线性矩阵方程 Hermitian正定解 特征值 解的存在性 

分 类 号：O15[理学—数学]