3 years ago

# On universal realizability of spectra

Ana I. Julio, Carlos Marijuán, Miriam Pisonero, Ricardo L. Soto

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 $Λ={λ1,λ2,…,λn}$ 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 ($UR$) if it is the spectrum of a nonnegative matrix for each possible Jordan canonical form allowed by Λ. It is well known that an $n×n$ 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 $ϵ≥0$ and $Λ={λ1,λ2,…,λn}$ is $UR,$ then ${λ1+ϵ,λ2,…,λn}$ is also $UR$. We give counter-examples for the cases: $Λ={λ1,λ2,…,λn}$ is $UR$ implies ${λ1+ϵ,λ2−ϵ,λ3,…,λn}$ is $UR,$ and $Λ1,Λ2$ are $UR$ implies $Λ1∪Λ2$ is $UR$.

DOI: S0024379518305366

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