On spectra realizable and diagonalizably realizable

Abstract

The list Λ={λ1,λ2,…,λn}, of complex numbers, is said to be realizable if it is the spectrum of an entrywise nonnegative matrix A. If A is diagonalizable, Λ is said to be diagonalizably realizable. Λ is universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ. Here, new realizability criteria are introduced. A diagonalizable version of a perturbation result by Rado is also proved. It allows to construct diagonalizable nonnegative matrices with prescribed spectrum. Criteria to decide the universal realizability of spectra are also established.