n×n矩阵空间的幂等基与对合基  

Idempotent Basis and Involutory Basis of n×n Matrix Space

作　　者：苏茹燕 杨忠鹏 陈梅香 Su Ruyan;Yang Zhongpeng;Chen Meixiang(School of Mathematics and Finance,Putian University,Putian 351100,China;College of Mathematics and Informatics,Fujian Normal University,Fuzhou 350001,China)

机构地区：[1]莆田学院数学与金融学院,福建莆田351100 [2]福建师范大学数学与信息学院,福建福州350001

出　　处：《北华大学学报（自然科学版）》2019年第6期708-713,共6页Journal of Beihua University（Natural Science）

基　　金：福建省自然科学基金项目(2018J01426);福建省教育科学“十三五”规划2019年度课题(2019CG0157);莆田学院教育教学改革研究项目(JG201915)

摘　　要：证明了n≥2时n×n阶矩阵空间存在无穷多个由幂等矩阵构成的幂等基,且对每个n×n阶矩阵给出了标准幂等基显示的线性表出系数.提出n×n阶矩阵空间基的基秩定义,证明了所有幂等基的基秩的最大下界和最小上界都是可达的.同时,首次得到n×n阶矩阵空间的对合基.When n≥2,there exists not only a basis of n ×n matrix space composed of idempotent matrices,but also infinitely many different idempotent bases.This leads to the conclusion that each n×n matrix can be uniquely linearly represented by idempotent matrices,and the linear expression coefficients are shown. It also gives the definition of base rank of basis of n×n matrix,and proves that the maximum lower bound and the minimum upper bound of the base rank of all idempotent basis are reachable.The involution base of n×n matrix space are obtained firstly.

关 键 词：矩阵空间 幂等基 对合基 基秩 线性表示 

分 类 号：O151.21[理学—数学]