关于矩阵的中心化子的刻画  

On the Characterization of the Centralizers of Matrices

作　　者：任芳国[1] 和嘉琪 周红军 REN Fangguo;HE Jiaqi;ZHOU Hongjun(School of Mathematics and Statistics,Shaanxi Normal University,Xi’an 710119,China)

机构地区：[1]陕西师范大学数学与统计学院,西安710119

出　　处：《大学数学》2022年第5期89-97,共9页College Mathematics

基　　金：国家自然科学基金资助项目(11471200);陕西本科和继续教育教学改革研究项目(21BY027)。

摘　　要：矩阵的普通乘法不满足交换律,而两个可换的矩阵具有许多良好的性质.本文通过矩阵的特征多项式、最小多项式、初等因子和Jordan块的特性,讨论了两个可换矩阵的特性,并给出了矩阵缠绕关系的性质,以此为基础探讨了矩阵A的中心化子C(A)与A的矩阵多项式集合P(A)之间的关系,获得了中心化子C(A)的维数和C(A)与P(A)相等的充分必要条件,旨在加深学生对可交换矩阵性质的理解和掌握,并促进学生学习高等(线性)代数的能力.The ordinary multiplication of matrices does not satisfy the commutative law,but two commutative matrices have many good properties.In this paper,the properties of two commutative matrices are discussed by the characteristic polynomials,minimum polynomials,elementary factors and Jordan block of matrices,and the properties of matrix entanglement are given.On this basis,the relationship between the centralizer of matrix and the set of matrix polynomials is discussed,the dimension of centralizer and the necessary and sufficient conditions for the equality of the centralizer and the set of matrix polynomials are obtained.The purpose is to deepen students′understanding and mastery of the properties of commutative matrix,and promote students′ability to learn advanced(linear)algebra.

关 键 词：矩阵可换 中心化子 缠绕关系 矩阵多项式 最小多项式 

分 类 号：O151.21[理学—数学]