Indices of diagonalizable and universal realizability of spectra

Abstract

A list $\Lambda =\{{\lambda }_1,\dots ,{\lambda }_n\}$ 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.

1. Theorem 2.4: we say that we are going to prove that there is a minimum called diagonalizable realizability index, but we do not. 2. Corollary 2.1: We say that we may define a universal realizablity index, but we do not show that such index exists. Corollary 2.1 only show lower and upper bounds for that index