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

doi:10.12677/PM.2023.1312393

期刊菜单

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 X^s-A₁^*X^-t₁A₁-A₂^*X^-t₂A₂=Q

DOI: 10.12677/PM.2023.1312393, PDF,
作者: 裴伟娟：河南开封科技传媒学院，河南开封
关键词: 非线性矩阵方程；Hermitian正定解；特征值；解的存在性；Nonlinear Matrix Equation； Hermitian Positive Definite Solution； Eigenvalue； Existence

摘要: 本文主要研究非线性矩阵方程X^s− 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。

Abstract: In this paper, we mainly investigate the Hermitian positive definite solution of the nonlinear matrix equation X^s− 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.

文章引用：裴伟娟. 关于非线性矩阵方程X^s-A₁^*X^-t₁A₁-A₂^*X^-t₂A₂=Q的若干结果[J]. 理论数学, 2023, 13(12): 3798-3802. https://doi.org/10.12677/PM.2023.1312393

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