矩阵线性组合幂等性及立方幂等性的一些结论  

Some results on idempotency and tripotency of linear combinations of matrices

作　　者：卜长江[1] 李娜[1] 孙艳玲[1] 

机构地区：[1]哈尔滨工程大学理学院,黑龙江哈尔滨150001

出　　处：《哈尔滨工程大学学报》2009年第12期1458-1460,共3页Journal of Harbin Engineering University

基　　金：黑龙江省自然科学基金资助项目(158110120002)

摘　　要：研究幂等矩阵和立方幂等矩阵的线性组合在矩阵理论和统计学中具有重要的意义.设A、B是2个n×n的复矩阵,令P=c1A+c2B,其中c1、c2为非零复数.该文在AB=BA的条件下分别给出:当A分别为幂等矩阵和立方幂等矩阵,B为任意矩阵时,线性组合P分别为幂等的和立方幂等的充分必要条件.并且利用以上结果直接得出下面的结论:当A为幂等矩阵,B为与A可交换的幂等矩阵或立方幂等矩阵时,P是幂等矩阵的充分必要条件;当A和B为可交换的立方幂等矩阵时,P是立方幂等矩阵的充分必要条件.It is important to consider the problem of linear combinations of idempotent and tripotent matrices in matrix theory and statistics. Let A and B be n x n nonzero complex matrices. Denote a linear combination of the two matrices by P = c1A ＋ c2B, where c1 and c2 are nonzero complex numbers. In this paper, if AB = BA, then we give （i） the sufficient and necessary conditions for idempotency of the matrix P, where A is an idempotent matrix and B is an arbitrary matrix ; （ii） the sufficient and necessary conditions for tripotency of the matrix P, where A is a tripotent matrix and B is an arbitrary matrix. Moreover, based on the above results, we have （i） the sufficient and necessary conditions for idempotency of the matrix P, where A is an idempotent matrix and B is a tripotent matrix or an idempotent matrix which is commutative with A ; （ii） the sufficient and necessary conditions for tripotency of the matrix P, where A and B are tripotent matrices that are commutative with each other.

关 键 词：幂等矩阵 立方幂等矩阵 交换性 线性组合 

分 类 号：O151.21[理学—数学] C81[理学—基础数学]