On universal realizability of spectra

A list Λ={λ1,λ2,…,λn} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be universally realizable (UR) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an n×n nonnegative matrix A is co-spectral to a nonnegative matrix B with constant row sums. In this paper, we extend the co-spectrality between A and B to a similarity between A and B, when the Perron eigenvalue is simple. We also show that if ϵ≥0 and Λ={λ1,λ2,…,λn} is UR, then {λ1+ϵ,λ2,…,λn} is also UR. We give counter-examples for the cases: Λ={λ1,λ2,…,λn} is UR implies {λ1+ϵ,λ2−ϵ,λ3,…,λn} is UR, and Λ1,Λ2 are UR implies Λ1∪Λ2 is UR.