Diagonalizably realizable implies universally realizable

Abstract

A spectrum Λ={λ1,…,λn} of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix A. The spectrum Λ is diagonalizably realizable (DR) if the realizing matrix A is diagonalizable, and Λ is universally realizable (UR) if it is realizable for each possible Jordan canonical form allowed by Λ. In 1981, Minc proved that if Λ is the spectrum of a diagonalizable positive matrix, then Λ is universally realizable. One of the main open questions about the problem of universal realizability of spectra is whether DR implies UR. Here, we prove a surprisingly simple result, which shows how diagonalizably realizable implies universally realizable.