On universal realizability of spectra

Abstract A list Λ = { λ 1 , λ 2 , … , λ n } 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 = { λ 1 , λ 2 , … , λ n } is UR , then { λ 1 + ϵ , λ 2 , … , λ n } is also UR . We give counter-examples for the cases: Λ = { λ 1 , λ 2 , … , λ n } is UR implies { λ 1 + ϵ , λ 2 − ϵ , λ 3 , … , λ n } is UR , and Λ 1 , Λ 2 are UR implies Λ 1 ∪ Λ 2 is UR .