Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method

Details Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method Potentially Nilpotent Patterns and the Nilpotent-Jacobian Method

2010-09-30

arxiv.org/abs/1010.0018v1

Mathematics

Rings and Algebras

12 pages

Scientific paper

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.

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