Minimum number of distinct eigenvalues allowed by a sign pattern

Abstract

This article initiates the study of the minimum number of distinct eigenvalues allowed by a sign pattern. Non-trivial examples of sign patterns that allow matrices with only one eigenvalue include the potentially nilpotent and the spectrally arbitrary sign patterns, in contrast to the inverse eigenvalue problem for graphs. Necessary digraph cycle conditions are developed for a sign pattern to have a matrix realization with exactly one eigenvalue. Certain manipulations of sign patterns provide new n×n patterns that preserve the minimum number of eigenvalues allowed, including some Jacobian methods. The 2×2 and the irreducible 3×3 sign patterns are classified according to the minimum number of eigenvalues allowed by the pattern.