Spectrally arbitrary ray patterns

Abstract

An n × n ray pattern A is said to be spectrally arbitrary if for every monic nth degree polynomial f ( x ) with coefficients from C , there is a matrix in the pattern class of A such that its characteristic polynomial is f ( x ) . In this article the authors extend the nilpotent-Jacobi method for sign patterns to ray patterns, establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary. They use this method to establish that a particular family of n × n irreducible ray patterns with exactly 3 n nonzeros is spectrally arbitrary. They then show that every n × n irreducible, spectrally arbitrary ray pattern has at least 3 n - 1 nonzeros.