矩阵线性组合幂等性及三次幂等性的研究  

SOME RESULTS ON IDEMPOTENCY AND TRIPOTENCY OF LINEAR COMBINATIONS OF MATRICES

作　　者：武玲玲 刘晓冀 Wu Lingling;Liu Xiaoji(College of Science,Guizhou University of Engineering Science,Bijie 551700;College of Science,Guangxi University for Nationalities,Nanning 530006)

机构地区：[1]贵州工程应用技术学院理学院,毕节551700 [2]广西民族大学理学院,南宁530006

出　　处：《高等学校计算数学学报》2021年第3期252-257,共6页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金(12061015);广西自然科学基金(2018GXNSFDA201023);广西十百千人才专工程项;广西高校高水平创新团队及卓越学者项目;贵州省教育厅青年科技人才成长项目(黔教合KY字[2019]157).

摘　　要：1引言及引理幂等矩阵和三次幂等矩阵的线性组合在矩阵理论和统计学中具有重要的应用[1,2].I_(n)表示C上的n×n单位矩阵,r=rank(A)表示A∈C^(n×n)的秩.设c_(1),c_(2)∈C_(1)是非零复数,A,B∈C^(n×n)是非零的复矩阵,且A≠±B,P是A和B的线性组合,即P=c_(1)A+c_(2)B.Let A and B be n×n nonzero complex matrices.Denote a linear combination of the two matrices by P=c_(1)A+c_(2)B,where c_(1),c_(2) are nonzero complex numbers.In this paper,if AB=BA,then we give the sufficient and necessary conditions for tripotency of the matrix P,where A is an idempotent matrix and B is an arbitrary matrix;and the sufficient and necessary conditions for idempotency of the matrix P,where A is a tripotent matrix and B is an arbitrary matrix.Moreover,based on the above results,we have the sufficient and necessary conditions for tripotent matrix P,where A is an idempotent matrix and B is an idempotency matrix or tripotent matrix which is commutative with A;and the sufficient and necessary conditions for idempotency matrix P,where A is a tripotent matrix and B is an idempotency matrix or tripotent matrix which is commutative with A.

关 键 词：线性组合 单位矩阵 幂等矩阵 矩阵理论 复矩阵 幂等性 

分 类 号：O151[理学—数学]