The NIEP for four dimensional Leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial
type: Item
instance of: scholarly article
title: The NIEP for four dimensional Leslie and doubly stochastic matrices with zero trace from the coefficients of the characteristic polynomial
zbMATH Open document ID: 1394.15024
author: Luca Benvenuti
published in: Linear Algebra and its Applications
publication date: 2018-06-12T00:00:00Z
review text: Let \(\Omega_n^0\) denote the set of all points in the complex plane that are eigenvalues of \(n\)-square doubly stochastic matrices with zero trace, and let \(\Pi_k^0\) be the union of the set \(\{ 1 \}\) and the convex hull of all \(k\)-th roots of unity not equal to \(1\). The author [Electron. J. Linear Algebra 30, 599--604 (2015; Zbl 1326.15022)] showed that the following inclusion holds: \[ \Omega_n^0 \supseteq \Pi_2^0 \cup \Pi_3^0 \cup \ldots \cup \Pi_{n-2}^0 \cup [-1,1] \cup \Pi_n^0.\eqno{(1)} \] Now, in the paper under review, the same author proves that, if \(n = 4,\) the equality sign holds in \((1)\). This is achieved by auxiliary results on the trace of powers and products of permutation matrices with zero trace, and by establishing constraints for the coefficients of the characteristic polynomial of an arbitrary 4-square doubly stochastic matrix with zero trace. Moreover, explicit counterexamples are given showing that equality cannot occur in \((1)\) if \(n \in \{ 5, 6 \}\). In the final part of the paper, a full characterization of the set of eigenvalues is obtained in the case of 4-square Leslie stochastic matrices with zero trace. (These consist of arbitrary entries \(a_{1j}, \, 2 \leq j \leq 4,\) such that \(a_{12}+a_{13}+a_{14} = 1, \; a_{21} = a_{32} = a_{43} = 1,\) and \(a_{ij} = 0\) otherwise.)
Mathematics Subject Classification ID: 15B51
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A29
zbMATH DE Number: 6883439
zbMATH Keywords: inverse eigenvalue problem
zbMATH Keywords: doubly stochastic matrix
zbMATH Keywords: Leslie stochastic matrix
MaRDI profile type: Publication
full work available at URL: j.laa.2018.01.016
OpenAlex ID: W2792331767
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Wikidata QID: Q130128728
DOI: 10.1016/J.LAA.2018.01.016
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description: scientific article; zbMATH DE number 6883439