Indices of diagonalizable and universal realizability of spectra

TL;DR

This paper investigates the conditions under which complex spectra can be realized by nonnegative matrices, focusing on diagonalizable and universal realizability, and introduces indices and methods to determine these properties.

Contribution

It introduces indices of realizability for diagonalizable and universal spectra and provides a new approach to assess universal realizability through spectrum merging techniques.

Findings

Established indices for diagonalizable and universal realizability.

Proved results enabling easier determination of universal realizability.

Developed spectrum merging methods to analyze realizability.

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.