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

作　　者：陈梅香 叶铃滢 杨忠鹏 CHEN Meixiang;YE Lingying;YANG Zhongpeng(School of Mathematics and Finance,Putian University,Putian 351100,Fujian Province,China;College of Mathematics and Informatics,Fujian Normal University,Fuzhou 350007,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,福州350007

出　　处：《吉林大学学报（理学版）》2021年第2期221-228,共8页Journal of Jilin University:Science Edition

基　　金：国家自然科学基金(批准号:61772292,11871404);福建省自然科学基金(批准号:2018J01426).

摘　　要：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.

关 键 词：广义二次矩阵 幂等矩阵 幂零矩阵 线性组合 非平凡解 

分 类 号：O151.21[理学—数学]