On new universal realizability criteria
type: Item
instance of: scholarly article
title: On new universal realizability criteria
zbMATH Open document ID: 1504.15062
author: Luis E. Arrieta
author: Ricardo L. Soto
published in: Special Matrices
publication date: 2022-12-08T00:00:00Z
review text: A list \(\Lambda = \{\lambda_1, \lambda_2, \dots, \lambda_n\}\) (a list is used instead of a set to handle multiplicities) of complex numbers is said to be \textit{realizable} if it is the spectrum (counting multiplicities) of a nonnegative matrix and is said to be universally realizable (UR), if it is realizable for each possible Jordan canonical form allowed by \(\Lambda\). In 1981, \textit{H. Minc} [Proc. Am. Math. Soc. 83, 665--669 (1981; Zbl 0472.15006)] proved that if \(\Lambda\) is diagonalizably positively realizable, then \(\Lambda\) is UR. The question whether this result holds for nonnegative realizations was open for almost 40 years. Two extensions of Mins's result have been obtained by \textit{M. Collao} et al. [Spec. Matrices 6, 301--309 (2018; Zbl 1404.15027)] and \textit{C. R. Johnson} et al. [Linear Algebra Appl. 587, 302--313 (2020; Zbl 1475.15012)]. In this paper, the authors exploit these extensions to generate new universal realizability criteria. Moreover, they also prove that under certain conditions, the union of two UR lists is also UR, and for certain criteria, if \(\Lambda\) is UR, then for \(t\ge 0\), \(\Lambda_t = \{\lambda_1 +t, \lambda_2 \pm t, \lambda_3, \dots, \lambda_n\}\) is also UR.
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15A20
zbMATH DE Number: 7629071
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: diagonalizable realizability
zbMATH Keywords: universal realizability
zbMATH Keywords: Jordan structure
reviewed by: Tin-Yau Tam
MaRDI profile type: Publication
full work available at URL: spma-2022-0177
OpenAlex ID: W4309616236
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DOI: 10.1515/SPMA-2022-0177
description: scientific article; zbMATH DE number 7629071