On Nilpotence of a Kind of Circulant Matrices over Zp

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.