On Nilpotence of a Kind of Circulant Matrices over Zp

TL;DR

This paper characterizes when certain circulant matrices over finite fields are nilpotent, providing necessary and sufficient conditions and a formula for their nilpotent index, thereby resolving a conjecture by C.Y. Zhang.

Contribution

It offers a complete characterization and formula for the nilpotent index of specific circulant matrices over Z_p, solving an open conjecture.

Findings

Derived necessary and sufficient conditions for nilpotence.

Established a formula for the nilpotent index.

Provided a complete solution to Zhang's conjecture.

Abstract

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.