Diagonalizably realizable implies universally realizable

Abstract

A spectrum $\Lambda=\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable if it is the spectrum of an entrywise nonnegative matrix $A$. The spectrum $\Lambda$ is diagonalizably realizable ($\mathcal{DR}$) if the realizing matrix $A$ is diagonalizable, and $\Lambda$ is universally realizable ($\mathcal{UR}$) if it is realizable for each possible Jordan canonical form allowed by $\Lambda.$ In 1981, Minc proved that if $\Lambda$ is the spectrum of a diagonalizable positive matrix, then $\Lambda$ is universally realizable. One of the main open questions about the problem of universal realizability of spectra is
whether $\mathcal{DR}$ implies $\mathcal{UR}$. Here, we prove a surprisingly simple result, which shows how diagonalizably realizable implies universally realizable.