Minimum number of distinct eigenvalues allowed by a sign pattern

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 …