On allowable properties and spectrally arbitrary sign pattern matrices

Abstract

In this paper, we give a geometric construction for different allowable properties for sign pattern matrices. In Section 2, we give a construction for detecting a sign pattern matrix to be potentially nilpotent and also compute the nilpotent matrix realization for a given sign pattern matrix if it exists. In Section 3, we develop a geometric construction for potential stability. In Section 4, we establish a necessary and sufficient condition for a sign pattern matrix S to be spectrally arbitrary. For a given sign pattern matrix of order n, we prove that there exists a surface of dimension at most m for m≤n such that for every vector a1,a2,⋯,an on the same surface, there exists a matrix A∈Q(S), a qualitative class of S whose characteristic polynomial is xn−a1xn−1+⋯+(−1)nan.