A counterexample on spectra of zero patterns

Abstract

An $n\times n$ zero pattern $S$, which is a matrix with entries $*$ and $0$, is called spectrally arbitrary with respect to a field $F$ if any monic polynomial $f$ of degree $n$ can be realized as the characteristic polynomial of a matrix obtained from $S$ by replacing the $*$'s with non-zero elements of $F$. We construct an $n\times n$ zero pattern that is spectrally arbitrary with respect to $\mathbb{C}$ and has $2n-1$ nonzero entries.