Universal realizability in low dimension

Abstract

We say that a list Λ = {λ1,...,λn} of complex numbers is realizable, if it is the spectrum of a nonnegative matrix A (a realizing matrix). We say that Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. This work studies the universal realizability of spectra in low dimension, that is, realizable spectra of size n ≤ 5. It is clear that for n ≤ 3 the concepts of universally realizable and realizable are equivalent. The case n = 4 is easily deduced from previous results in [7]. We characterize the universal realizability of real spectra of size 5 and trace zero, and we describe a region for the universal realizability of nonreal 5-spectra with trace zero. As an important by-product of our study, we also show that realizable lists on the left half- plane, that is, lists Λ = {λ1, . . . , λn}, where λ1 is the Perron eigenvalue and Re λi ≤ 0, for i = 2,...,n, are not necessarily universally realizable.