m次数量幂等矩阵线性组合的可逆性

Invertiblity of Linear Combinations of m-scalar-idempotent Matrices

如果有非零数λ与μ使P^m=λP,Q^m=μQ,则称P,Q分别是由λ,μ确定的m次数量幂等矩阵.本文证明了,若有非零数a与b,当λa^(m-1)-(-1)^(m-1)μb^(m-1)≠0时,使可交换的分别由λ,μ确定的m次数量幂等矩阵P,Q的线性组合aP+bQ是可逆的,那么对任意非零数u,v,当λu^(m-1)-(-1)^(m-1)μv^(m-1)≠0时,uP+vQ也是可逆的.本文主要结果和方法的应用,可以推广已有文献的2次、3次幂等矩阵的线性组合可逆的结论.

We call the matrix P, Q as m-scalar-idempotent matrices determined by λ,μ respectively, if there exist nonzero numbers ）λ,μ, such that Pm = λP , Qm = μQ. This paper proves that, if linear combination aP＋bQ of commute m-scalar-idempotent matricesP, Q deter- mined by λ, μ respectively is invertible, when λμm-1-（-1）m-1μvm-1≠0 for arbitrary nonzero numbers a, b, then uP ＋ vQ is also invertible, if λμm-1-（-1）m-1μvm-1≠0, in which u, v are nonzero. As an application of main results and methods used in the paper, it can generalize the conclusions about the invertibility of linear combinations of idempotent and tripotent matrices in present literatures.