Bounds on the bases of irreducible generalized sign pattern matrices

Li et al. [Z. Li, F. Hall, C. Eschenbach, On the period and base of a sign pattern matrix, Linear Algebra Appl. 212/213 (1994) 101–120] extended the concept of the base (or “index of convergence”) and period from nonnegative matrices to powerful sign pattern matrices. In this paper we study the bases for non-powerful irreducible sign pattern matrices (and more generally, for generalized sign pattern matrices). We obtain sharp upper bounds, together with a complete characterization of the equality cases, of the bases for both primitive and irreducible sign pattern (and generalized sign pattern) matrices. We also show that there exist “gaps” in the base set of the classes of such matrices.