Bull. Korean Math. Soc. 2019; 56(2): 305-318
Online first article March 8, 2019 Printed March 1, 2019
https://doi.org/10.4134/BKMS.b171108
Copyright © The Korean Mathematical Society.
Ber-Lin Yu
Huaiyin Institute of Technology
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.
Keywords: eventually positive matrix, broom sign pattern, checkerboard block sign pattern
MSC numbers: 15A48, 15A18, 05C50
Supported by: The author’s work are supported by the Natural Science Foundation of HYIT under grant number 16HGZ007
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