Sparse inertially arbitrary patterns

An n-by-n sign pattern A is a matrix with entries in {+,-,0}. An n-by-n nonzero pattern A is a matrix with entries in {∗,0} where ∗ represents a nonzero entry. A pattern A is inertially arbitrary if for every set of nonnegative integers n1,n2,n3 with n1+n2+n3=n there is a real matrix with pattern A having inertia (n1,n2,n3). We explore how the inertia of a matrix relates to the signs of the coefficients of its characteristic polynomial and describe the inertias allowed by certain sets of polynomials. This information is useful for describing the inertia of a pattern and can help show a pattern is inertially arbitrary. Britz et al. [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary sign patterns, SIAM J. Matrix Anal. Appl. 26 (2004) 257–271] conjectured that irreducible spectrally arbitrary patterns must have at least 2n nonzero entries; we demonstrate that irreducible inertially arbitrary patterns can have less than 2n nonzero entries.