Zero-nonzero patterns for nilpotent matrices over finite fields Academic Article uri icon

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abstract

  • Fix a field F. A zero-nonzero pattern A is said to be potentially nilpotent over F if there exists a matrix with entries in F with zero-nonzero pattern A that allows nilpotence. In this paper we initiate an investigation into which zero-nonzero patterns are potentially nilpotent over F, with a special emphasis on the case that F = Z_p is a finite field. As part of this investigation, we develop methods, using the tools of algebraic geometry and commutative algebra, to eliminate zero-nonzero patterns A as being potentially nilpotent over any field F. We then use these techniques to classify all irreducible zero-nonzero patterns of order two and three that are potentially nilpotent over Z_p for each prime p.

publication date

  • October 1, 2009