Indices of diagonalizable and universal realizability of spectra

Abstract

A list $\Lambda =\{\lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers (repeats allowed) is said to be \textit{realizable} if it is the spectrum of an entrywise nonnegative matrix $A$. $\Lambda $ is \textit{diagonalizably realizable} if the realizing matrix $A$ is diagonalizable. $\Lambda $ is said to be \textit{universally realizable} if it is \textit{\ realizable} for each possible Jordan canonical form allowed by $\Lambda .$ Here, we study the connection between diagonalizable realizability and universal realizability of spectra. In particular, we establish \textit{\ indices of realizability} for diagonalizable and universal realizability. We also define the merge of two spectra and we prove a result that allow us to easily decide, in many cases, about the universal realizability of spectra.