A counterexample on spectra of zero patterns

Abstract

An $n\times n$ zero pattern $S$, which is a matrix with entries $\ast$ 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 $\ast$'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.

3 pages, a bit stronger result added

References in corpus (1)

• Counterexamples on spectra of sign patterns

Cited by in corpus (1)

• Three observations on spectra of zero-nonzero patterns