Potentially nilpotent patterns and the Nilpotent-Jacobian method

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 of Pereira [Nilpotent matrices and spectrally arbitrary sign patterns. Electron. J. Linear Algebra 16 (2007) 235], we 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 arbitrary pattern, then N must have full index.