Potentially nilpotent patterns and the Nilpotent-Jacobian method
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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.
