4 x 4 Irreducible sign pattern matrices that require four distinct eigenvalues

A sign pattern matrix is a matrix whose entries are from the set {+, -, 0}. For a sign pattern matrix A, the qualitative class of A, denoted Q(A), is the set of all real matrices whose entries have signs given by the corresponding entries of A. An n x n sign pattern matrix A requires all distinct eigenvalues if every real matrix in Q(A) has n distinct eigenvalues. Li and Harris (2002) [13] characterized the 2 x 2 and 3 x 3 irreducible sign pattern matrices that require all distinct eigenvalues, and established some useful general results on n x n sign patterns that require all distinct eigenvalues. In this paper, we characterize 4 x 4 irreducible sign patterns that require four distinct eigenvalues. This is done by characterizing 4 x 4 irreducible sign patterns that require four distinct real eigenvalues, that require four distinct nonreal real eigenvalues, or that require two distinct real eigenvalues and a pair of conjugate nonreal eigenvalues. The last case turns out to be much more involved. Some interesting open problems are presented. Three important tools that are used in the paper are the following: the discriminant of a polynomial; the fact that if a square sign pattern matrix A requires all distinct eigenvalues then A requires a fixed number of real eigenvalues; and the known result that if A is a "k-cycle" sign pattern then for each B E Q(A), the k nonzero eigenvalues of B are evenly distributed on a circle in the complex plane centered at the origin.(c) 2023 Elsevier Inc. All rights reserved.