Spectra inhabiting the left half-plane that are universally realizable
type: Item
instance of: scholarly article
title: Spectra inhabiting the left half-plane that are universally realizable
zbMATH Open document ID: 1485.15018
author: Ricardo L. Soto
published in: Special Matrices
publication date: 2022-02-18T00:00:00Z
review text: A list \(\Lambda :=\left\{ \lambda _{1},\dots,\lambda _{n}\right\} \) of complex numbers is called realizable if it is the spectrum of an \(n\times n\) nonnegative matrix \(A.\) Since a nonnegative matrix has a dominant eigenvalue, we restrict to the case where \(\lambda _{1}\) is real and positive and \(\lambda _{1}\geq \left\vert \lambda _{j}\right\vert \) for all \( j\geq 2\). Moreover, \(\Lambda \) is universally realizable (UR) if in addition \(A\) can be chosen to have any of the Jordan forms allowable by \(\Lambda \). A classical theorem of \textit{H. Minc} [Proc. Am. Math. Soc. 83, 665--669 (1981; Zbl 0472.15006)] shows that if there is a diagonalizable matrix with strictly positive entries which realizes \(\Lambda \) then \(\Lambda \) is UR. Pursuing the efforts which have been made to to relax the conditions in Minc's theorem, the author examines the case where all \(\lambda _{j}\) (except for \(\lambda _{1}\)) lie in the left half-plane and provides new criteria for \(\Lambda \) to be UR. New examples are provided of \(\Lambda \) which are UR but not covered by previous theorems.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A20
zbMATH DE Number: 7476488
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: universal realizability
zbMATH Keywords: Jordan structure
MaRDI profile type: Publication
full work available at URL: spma-2021-0155
OpenAlex ID: W4205904618
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DOI: 10.1515/SPMA-2021-0155
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description: scientific article; zbMATH DE number 7476488