${\mathcal{D}}_{n,r}$ is not potentially nilpotent for $n\ge 4r-2$

Shao, Yanling ; Gao, Yubin ; Gao, Wei

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 3, pp. 671-679.

Résumé

An $n\times n$ sign pattern $\mathcal{A}$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $\mathcal{A}$. Let ${\mathcal{D}}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$ $(i=1,\dots ,r)$ and $(i,i-r+1)$ $(i=r+1,\dots ,n)$ entries are negative, and zeros elsewhere. We prove that for $r\ge 3$ and $n\ge 4r-2$, the sign pattern ${\mathcal{D}}_{n,r}$ is not potentially nilpotent, and so not spectrally arbitrary.

Détail

DOI : 10.1007/s10587-016-0285-7

Classification : 05C50, 15A18
Mots-clés : sign pattern; potentially nilpotent pattern; spectrally arbitrary pattern