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sign pattern; potentially nilpotent pattern; spectrally arbitrary pattern
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 $2\leq r \leq n$ 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 $r\geq 3$ and $n \geq 4r-2$, the sign pattern $\mathcal {D}_{n,r}$ is not potentially nilpotent, and so not spectrally arbitrary.
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