Universal realizability in low dimension
type: Item
instance of: scholarly article
title: Universal realizability in low dimension
zbMATH Open document ID: 1473.15016
author: Ana I. Julio
author: Ricardo L. Soto
author: C. Marijuán
author: M. Pisonero
published in: Linear Algebra and its Applications
publication date: 2021-04-24T00:00:00Z
review text: This paper deals with the nonnegative inverse eigenvalue problem (NIEP) for matrices of size \(n \times n\), \(n \leq 5\). The aim is to find necessary and sufficient conditions for a list \(\Lambda=\{\lambda_1,\dots,\lambda_n\}\) of complex numbers to be the spectrum of a nonnegative matrix \(A\). \(\Lambda\) is said to be realizable and \(A\) is a realizing matrix. A realizable list \(\Lambda\) is called universally realizable (UR) if, for every possible Jordan canonical form allowed by \(\Lambda\), there is a nonnegative matrix with spectrum \(\Lambda\). The problem of finding necessary and sufficient conditions for a realizable list \(\Lambda\) to be universally realizable is called universal realizability problem (URP). It is clear that for \(n \leq 3\) the concepts of universally realizable and realizable are equivalent. The case \(n=4\) is easily deduced from previous results of the authors. The URP has been completely solved, with different approaches, for different types of lists. Here, the authors characterize the universal realizability of real spectra of size 5 and trace zero by applying several criteria and results from the NIEP. In addition, they describe a region for the universal realizability of non-real 5-spectra with trace zero. The authors distinguish two cases: (a) non-real spectra with repeated complex eigenvalues, and (b) non-real spectra with repeated real eigenvalues. To prove the nonexistence of certain Jordan canonical forms, the authors use methods based on graph theory. Finally, the authors show that realizable lists on the left half-plane, that is, lists \(\Lambda=\{\lambda_1,\dots,\lambda_n\}\), where \(\lambda_1\) is the Perron eigenvalue and \(\mathrm{Re} (\lambda_i) \leq 0\), for \(i=1,2,\ldots,n\), are not necessarily universally realizable.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A29
zbMATH DE Number: 7337640
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: inverse eigenvalue problem
zbMATH Keywords: universal realizability
zbMATH Keywords: Jordan canonical form
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2021.02.012
OpenAlex ID: W3131295367
cites work: Eigenvalues and eigenvectors of symmetric centrosymmetric matrices
cites work: Universal realizability of spectra with two positive eigenvalues
cites work: Spectra universally realizable by doubly stochastic matrices
cites work: Nonnegative inverse elementary divisors problem in the left half plane
cites work: Nonnegative realizability with Jordan structure
cites work: On universal realizability of spectra
cites work: A characterization of trace zero nonnegative \(5\times 5\) matrices
cites work: Inverse Elementary Divisor Problem for Nonnegative Matrices
cites work: Q5510191
cites work: The inverse eigenvalue problem for nonnegative matrices
cites work: Nonnegative matrices with prescribed elementary divisors
cites work: Nonnegative matrices with prescribed spectrum and elementary divisors
cites work: Applications of a Brauer theorem in the nonnegative inverse eigenvalue problem
cites work: A characterization of trace zero symmetric nonnegative \(5\times 5\) matrices
cites work: The nonnegative inverse eigenvalue problem from the coefficients of the characteristic polynomial. EBL digraphs
DOI: 10.1016/J.LAA.2021.02.012
description: scientific article; zbMATH DE number 7337640