# On universal realizability of spectra

Publication date: Available online 9 November 2018

**Source:** Linear Algebra and its Applications

Author(s): Ana I. Julio, Carlos Marijuán, Miriam Pisonero, Ricardo L. Soto

##### Abstract

A list $\mathrm{\Lambda}=\{{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n}\}$ of complex numbers is said to be *realizable* if it is the spectrum of an entrywise nonnegative matrix. The list Λ is said to be *universally realizable* ($\mathcal{UR}$) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an $n\times n$ nonnegative matrix *A* is co-spectral to a nonnegative matrix *B* with constant row sums. In this paper, we extend the co-spectrality between *A* and *B* to a similarity between *A* and *B*, when the Perron eigenvalue is simple. We also show that if $\u03f5\ge 0$ and $\mathrm{\Lambda}=\{{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n}\}$ is $\mathcal{UR},$ then $\{{\lambda}_{1}+\u03f5,{\lambda}_{2},\dots ,{\lambda}_{n}\}$ is also $\mathcal{UR}$. We give counter-examples for the cases: $\mathrm{\Lambda}=\{{\lambda}_{1},{\lambda}_{2},\dots ,{\lambda}_{n}\}$ is $\mathcal{UR}$ implies $\{{\lambda}_{1}+\u03f5,{\lambda}_{2}-\u03f5,{\lambda}_{3},\dots ,{\lambda}_{n}\}$ is $\mathcal{UR},$ and ${\mathrm{\Lambda}}_{1},{\mathrm{\Lambda}}_{2}$ are $\mathcal{UR}$ implies ${\mathrm{\Lambda}}_{1}\cup {\mathrm{\Lambda}}_{2}$ is $\mathcal{UR}$.

Publisher URL: https://www.sciencedirect.com/science/article/pii/S0024379518305366

DOI: S0024379518305366

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