Potentially nilpotent and spectrally arbitrary even cycle sign patterns

Abstract

An n × n sign pattern S n is potentially nilpotent if there is a real matrix having sign pattern S n and characteristic polynomial x n . A new family of sign patterns C n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern C n . These nilpotent matrices are used together with a Jacobian argument to show that C n is spectrally arbitrary, i.e., there is a real matrix having sign pattern C n and characteristic polynomial x n + ∑ i = 1 n ( - 1 ) i μ i x n - i for any real μ i . Some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given.