On universal realizability of spectra

Abstract

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 ǫ ≥ 0 and Λ = {λ1, λ2, . . . , λn} is UR, then {λ1 + ǫ, λ2, . . . , λn} is also UR. We give counter-examples for the cases: Λ = {λ1, λ2, . . . , λn} is UR implies {λ1 + ǫ, λ2 − ǫ, λ3, . . . , λn} is UR, and Λ1,Λ2 are UR implies Λ1 ∪ Λ2 is UR.