Efficient realization of nonzero spectra by polynomial matrices

Abstract

A theorem of Boyle and Handelman gives necessary and sufficient conditions for an $n$-tuple of nonzero complex numbers to be the nonzero spectrum of some matrix with nonnegative entries, but is not constructive and puts no bound on the necessary dimension of the matrix. Working with polynomial matrices, we constructively reprove this theorem in a special case, with a bound on the size of the polynomial matrix required to realize a given polynomial.