Notes on the 2n-conjecture∗

A sign pattern is an n × n matrix, A, with entries in {+,−, 0}. If A is a real n× n matrix for which each entry has the same sign as its corresponding entry in A, then A is a realization of A, and we write A ∈ A. The 2n-conjecture is related to the study of the spectral properties among the matrices in A. The n × n sign pattern A is a spectrally arbitrary pattern (or a SAP, for short), provided that for each real, monic polynomial r(x) of degree n there is a realization A ∈ A whose characteristic polynomial pA(x) is r(x). Equivalently, A is a SAP provided each conjugate-closed multi-set of n complex numbers is the spectrum of at least one realization of A. As an example, consider the sign-pattern