Potentially eventually positive broom sign patterns

Bull. Korean Math. Soc. Published online 2019 Mar 08

Yu Ber-Lin
Huaiyin Institute of Technology

Abstract : A sign pattern is a matrix whose entries belong to the set $\{+, -, 0\}$. An $n$-by-$n$ sign pattern $\mathcal{A}$ is said to allow an eventually positive matrix or be potentially eventually positive if there exist at least one real matrix $A$ with the same sign pattern as $\mathcal{A}$ and a positive integer $k_{0}$ such that $A^{k}>0$ for all $k\geq k_{0}$. Identifying the necessary and sufficient conditions for an $n$-by-$n$ sign pattern to be potentially eventually positive, and classifying the $n$-by-$n$ sign patterns that allow an eventually positive matrix are two open problems. In this article, we focus on the potential eventual positivity of broom sign patterns. We identify all the minimal potentially eventually positive broom sign patterns. Consequently, we classify all the potentially eventually positive broom sign patterns.