Potentially nilpotent tridiagonal sign patterns of order 4

Abstract

An $n\times n$ sign pattern ${\cal A}$ is said to be potentially nilpotent (PN) if there exists some nilpotent real matrix $B$ with sign pattern ${\cal A}$. In [M.~Arav, F.~Hall, Z.~Li, K.~Kaphle, and N.~Manzagol.Spectrally arbitrary tree sign patterns of order $4$, {\em Electronic Journal of Linear Algebra}, 20:180--197, 2010.], the authors gave some open questions, and one of them is the following: {\em For the class of $4 \times 4$ tridiagonal sign patterns, is PN (together with positive and negative diagonal entries) equivalent to being SAP?}\ In this paper, a positive answer for this question is given.