A Class of Minimally Spectrally Arbitrary Sign Patterns

Abstract

A spectrally arbitrary pattern \({A}\) is a sign pattern of order \(n\) such that every monic real polynomial of degree \(n\) can be achieved as the characteristic polynomial of a matrix with sign pattern \({A}\). A sign pattern \({A}\) is minimally spectrally arbitrary if it is spectrally arbitrary but is not spectrally arbitrary if any nonzero entry (or entries) of \({A}\) is replaced by zero. In this paper, we introduce some new sign patterns which are minimally spectrally arbitrary for all orders \(n\geq 7\).