Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method

TL;DR

This paper investigates potentially nilpotent matrix patterns, introduces construction methods including balanced trees, and analyzes the spectral properties and limitations of the Nilpotent-Jacobian method in establishing spectral arbitrariness.

Contribution

It develops new techniques for constructing potentially nilpotent patterns and reveals a necessary condition for the Nilpotent-Jacobian method to prove spectral arbitrariness.

Findings

Balanced tree patterns can be potentially nilpotent.

Some balanced trees are spectrally arbitrary via the Nilpotent-Jacobian method.

Nilpotent matrices used in this method must have full index.

Abstract

A nonzero pattern is a matrix with entries in {0,*}. A pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern. We develop ways to construct some potentially nilpotent patterns, including some balanced tree patterns. We explore the index of some of the nilpotent matrices constructed,and observe that some of the balanced trees are spectrally arbitrary using the Nilpotent-Jacobian method. Inspired by an argument in [R. Pereira, Nilpotent matrices and spectrally arbitrary sign patterns. Electron. J. Linear Algebra, 16 (2007), 232--236], we also uncover a feature of the Nilpotent-Jacobian method. In particular, we show that if N is the nilpotent matrix employed in this method to show that a pattern is a spectrally arbitary pattern, then N must have full index.