Negativity compensation in the nonnegative inverse eigenvalue problem
type: Item
instance of: scholarly article
title: Negativity compensation in the nonnegative inverse eigenvalue problem
zbMATH Open document ID: 1079.15503
author: Alberto Borobia
author: Julio Moro
author: Ricardo L. Soto
published in: Linear Algebra and its Applications
publication date: 2005-01-11T00:00:00Z
full work available at URL: http://hdl.handle.net/10533/175014
review text: A set \(\Lambda \) of complex numbers is \textit{realizable} if it is the spectrum of an entrywise nonnegative matrix. \textit{R. L. Soto} [Linear Algebra Appl. 369, 169--184 (2003; Zbl 1031.15018)] proved that it is possible a real set \(\Lambda =\Lambda _{1}\cup \dots \cup \Lambda _{k}\) be realizable, without all the sets \(\Lambda _{1},\dots ,\Lambda _{k}\) be realizable. In the paper under review, the authors extend this result, allowing the sets to be complex. The main tool is a result due to \textit{A. Brauer} [Duke Math. J. 19, 75--91 (1952; Zbl 0046.01202); Theorem 3.1] then they associate to any nonrealizable set \(\Gamma \) a certain negativity \({\mathcal N}(\Gamma )\) and to any realizable set \(\Gamma \) a certain negativity \({\mathcal M}(\Gamma )\). A result of \textit{Kh. R. Suleĭmanova} [Dokl. Akad. Nauk SSSR, n. Ser. 66, 343--345 (1949; Zbl 0035.20903)] is also generalized, considering the complex set \(\{z\in \mathbb C :Re\, z\leq 0\), \(| Re\, z| \geq | Im\, z| \}\) that includes the negative real semi-axis. Some examples are given in the real and complex cases.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 2126343
zbMATH Keywords: inverse eigenvalue problem
zbMATH Keywords: nonnegative matrices
zbMATH Keywords: constant row sum
zbMATH Keywords: Brauer's theorem
MaRDI profile type: Publication
OpenAlex ID: W2019562133
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DOI: 10.1016/J.LAA.2003.10.023
description: scientific article; zbMATH DE number 2126343