On potentially nilpotent double star sign patterns

Abstract

A matrix Open image in new window whose entries come from the set {+, −, 0} is called a sign pattern matrix, or sign pattern. A sign pattern is said to be potentially nilpotent if it has a nilpotent realization. In this paper, the characterization problem for some potentially nilpotent double star sign patterns is discussed. A class of double star sign patterns, denoted by \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(m, 2), is introduced. We determine all potentially nilpotent sign patterns in \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(3, 2) and \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(5, 2), and prove that one sign pattern in \( \mathcal{D}\mathcal{S}\mathcal{S}\mathcal{P} \)(3, 2) is potentially stable.