广义二次矩阵与其幂等矩阵线性组合幂等性的非平凡解

Nontrivial Solutions of Idempotency of Linear Combinations ofGeneralized Quadratic Matrix and Its Idempotent Matrix

首先,用广义二次矩阵的基本性质,研究表示为A^(2)=αA+βP的广义二次矩阵A与幂等矩阵P的线性组合ρA+σP为幂等的非平凡解(ρ,σ)的存在性,结果表明,当η^(2)=4β+α^(2)≠0时,ρA+σP有且仅有两个非平凡解,A可唯一地表示为这两个非平凡解生成的幂等矩阵的线性组合;其次,讨论当η^(2)=4β+α^(2)=0时ρA+σP非平凡解的情况.

Firstly,by using the basic properties of generalized quadratic matrices,we studied the existence of the nontrivial solution(ρ,σ)for the linear combinationρA+σP of generalized quadratic matrix A and an idempotent matrix P expressed as A^(2)=αA+βP.The results show that whenη^(2)=4β+α^(2)≠0,ρA+σP has only two nontrivial solutions,and the matrix A can be uniquely expressed as a linear combination of idempotent matrix generated by these two nontrivial solutions.Secondly,we discussed the case for nontrivial solution ofρA+σP whenη^(2)=4β+α^(2)=0.