Sign pattern matrices that allow inertia $\mathbb{S}_{n}$

Abstract

Sign pattern matrices of order $n$ that allow inertias in the set ${\mathbb{S}}_n$ are considered. All sign patterns of order 3 (up to equivalence) that allow ${\mathbb{S}}_3$ are classified and organized according to their associated directed graphs. Furthermore, a minimal set of such matrices is found. Then, given a pattern of order $n$ that allows ${\mathbb{S}}_n$, a construction is given that generates families of irreducible sign patterns of order $n+1$ that allow ${\mathbb{S}}_{n+1}$.