一类矩阵线性组合的对合性

Involutiveness of a linear combination of a class matrix

设A2和A2是2个n×n的非零复矩阵,矩阵A为A1、A2的线性组合,即A=c1A1+c2A2,其中c1、c2为非0复数.对矩阵线性组合的幂等性、立方幂等性以及对合性的研究在很多领域都有着重要的应用.利用立方幂等矩阵的标准型,且在A1、A2无交换性条件下,给出了当A1为幂等矩阵,A2为立方幂等矩阵时,它们的线性组合A是对合矩阵的充分必要条件.

Let A1 and A2 be nonzero complex matrices.Denote a linear combination of the two matrices by A=c1A1＋c2A2,where c1,c2∈C are nonzero scalars.The study of idemopotency,tripotency and involution of linear combinations of matrices has important applications in many fields.In this paper,by use of the standard form of a tripotent matrix,we characterized the sufficient and necessary conditions for the involutive combination of A,with A1,A2 as idempotent and tripotent matrices,respectively,without the assumption that A1A2=A2A1.