Potential Eventual Positivity of One Specific Tree Sign Pattern

摘要

A sign pattern is a matrix whose entries belong to the set {+, -, 0}. An n-by-n sign pattern A is said to allow an eventually positive matrix if there exist some real matrices A with the same sign pattern as A and a positive integer k_0 such that A~k > 0 for all k ≥ k_0. Identifying and classifying the n-by-n sign patterns that allow an eventually positive matrix were posed as two open problems by Berman, Catral, Dealba, et al. [Sign patterns that allow eventual positivity, ELA, 19(2010): 108-120]. In this article, a tree sign pattern A obtained from the double star sign pattern S_(3,2) by adding an pendent edge is investigated. Some necessary or sufficient conditions for the sign pattern A to allow an eventually positive matrix are established first. Then all the minimal tree sign patterns that allow an eventually positive matrix are identified as five specific tree sign patterns. Finally, all the tree sign patterns that allow an eventually positive matrix is classified.