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

@article{Shao2016,
abstract = {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.},
author = {Shao, Yan Ling, Gao, Yubin, Gao, Wei},
journal = {Czechoslovak Mathematical Journal},
keywords = {sign pattern; potentially nilpotent pattern; spectrally arbitrary pattern},
language = {eng},
number = {3},
pages = {671-679},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {$\mathcal \{D\}_\{n,r\}$ is not potentially nilpotent for $n \ge 4r-2$},
url = {http://eudml.org/doc/286814},
volume = {66},
year = {2016},
}

TY - JOUR
AU - Shao, Yan Ling
AU - Gao, Yubin
AU - Gao, Wei
TI - $\mathcal {D}_{n,r}$ is not potentially nilpotent for $n \ge 4r-2$
JO - Czechoslovak Mathematical Journal
PY - 2016
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 3
SP - 671
EP - 679
AB - 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.
LA - eng
KW - sign pattern; potentially nilpotent pattern; spectrally arbitrary pattern
UR - http://eudml.org/doc/286814
ER -