关于Zp上一类循环矩阵的幂零性
On Nilpotence of a Kind of Circulant Matrices over Zp
摘要：
本文研究了域$Z_p$上一类循环矩阵$T_{n,m}$的幂零性，其中$T_{n,m}= \sum_{i = 0}^{m - 1} {S_n^i}$和$S_n$是$n$阶的基本循环矩阵.给出了关于$n$和$m$判定$T_{n,m}$在$Z_p$上为零的充要条件.此外，当满足该条件时，我们还得到了$T_{n,m}$的幂零指数公式。作为应用，我们给出了一个猜想的完全解.
英文摘要：
We investigate the nilpotence of a kind of circulant matrices $T_{n,m}$ over field $Z_p$ where $T_{n,m}= \sum_{i = 0}^{m - 1} {S_n^i}$ and $S_n$ is the fundamental circulant matrix of order $n$. The necessary and sufficient condition on $n$ and $m$ for determining nilpotence of $T_{n,m}$ over $Z_p$ is presented. Moreover, we obtain a formula for nilpotent index of $T_{n,m}$ when the condition is satisfied. As an application, we give a complete solution of a conjecture of C.Y.Zhang.
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