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

• Volume: 66, Issue: 3, page 671-679
• ISSN: 0011-4642

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## Abstract

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

## How to cite

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Shao, Yan Ling, Gao, Yubin, and Gao, Wei. "$\mathcal {D}_{n,r}$ is not potentially nilpotent for $n \ge 4r-2$." Czechoslovak Mathematical Journal 66.3 (2016): 671-679. <http://eudml.org/doc/286814>.

@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 -

## References

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1. Brualdi, R. A., Ryser, H. J., Combinatorial Matrix Theory, Encyclopedia of Mathematics and Its Applications 39 Cambridge University Press, Cambridge (1991). (1991) Zbl0746.05002MR1130611
2. Catral, M., Olesky, D. D., Driessche, P. van den, Allow problems concerning spectral properties of sign pattern matrices: a survey, Linear Algebra Appl. 430 (2009), 3080-3094. (2009) MR2517861
3. Cavers, M. S., Meulen, K. N. Vander, Spectrally and inertially arbitrary sign patterns, Linear Algebra Appl. 394 (2005), 53-72. (2005) MR2100576
4. Gao, Y., Li, Z., Shao, Y., A note on spectrally arbitrary sign patterns, JP J. Algebra Number Theory Appl. 11 (2008), 15-35. (2008) Zbl1163.15008MR2458665
5. Garnett, C., Shader, B. L., A proof of the ${T}_{n}$ conjecture: Centralizers, Jacobians and spectrally arbitrary sign patterns, Linear Algebra Appl. 436 (2012), 4451-4458. (2012) Zbl1244.15020MR2917422
6. Horn, R. A., Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge (1985). (1985) Zbl0576.15001MR0832183

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