数量对合矩阵的线性组合的秩的不变性  

Invariance of Rank for Linear Combinations of Scalar-involutory Matrices

作　　者：张金辉[1] 王海明[1] 杨忠鹏[1] 胡清孝[1,2] 

机构地区：[1]莆田学院数学系,福建莆田351100 [2]福建师范大学数计学院,福建福州350007

出　　处：《数学研究》2010年第1期98-103,共6页Journal of Mathematical Study

基　　金：2008年福建省高校服务海西建设重点项目(2008HX03);福建省教育厅科研资助项目(JA08196);莆田学院科研资助项目(2006Z006)

摘　　要：若矩阵A、B满足A^2=λ~2I、B^2=μ~2I(λμ≠0),称A、B都是数量对合矩阵.当非零复数a、b、u、v满足aλ+bμ≠0、μλ+vμ≠0时,我们证明了数量对合矩阵A、B与单位矩阵Ⅰ的线性组合的秩总是相等,并且是一个与a、b、u,v选择都无关的常数.应用所得到数量对合矩阵的线性组合的秩的不变性,可推广已有文献的关于对合矩阵的相应结果.The square matrices A and B over the complex field C are said to be scalar-involutory matrices if A2 = λ2I and B2 = μ2I, where λμ≠ 0. When all nonzero scalars a, b, u, v∈ C satisfying aλ ＋ bμ≠ 0 and u）, ＋ uλ＋vμ≠0, we proved the rank for the linear combination of a pair of scalar-involutory matrices A, B and a identity matrix I is constant and independent on the choice of a, b, u and v. Then the corresponding results of involutory matrices in previous literatures are generalized by the invariance of rank for linear combinations of scalar-involutory matrices.

关 键 词：数量对合矩阵 线性组合 秩等式 秩不变性 

分 类 号：O151.21[理学—数学]