The nilpotent-centralizer method for spectrally arbitrary patterns

This paper introduces a new criterion for an n×n sign-pattern (respectively, zero–nonzero pattern), A to be spectrally arbitrary; that is to have the property that for each monic real polynomial r(x) of degree n there exists a matrix with sign-pattern (respectively, zero–nonzero pattern) A that has r(x) as its characteristic polynomial. To date, analytic properties of a certain polynomial map associated with A have been used to prove that A is spectrally arbitrary. In this paper, we derive a method that uses the algebraic structure of a certain nilpotent matrix to show that a sign-pattern (respectively, zero–nonzero pattern) is spectrally arbitrary.