On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral
type: Item
instance of: scholarly article
title: On the Perron-Frobenius eigenvector for nonnegative integral matrices whose largest eigenvalue is integral
zbMATH Open document ID: 0622.15012
DOI: 10.1016/0024-3795(87)90081-4
author: Jonathan J. Ashley
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let A be a k by k nonnegative matrix with spectral radius r. For \(\lambda\geq r\) it is shown that \(| \det (\lambda I-A)| \leq \lambda^ k-r^ k\). The proof proceeds by induction on k using an unusual formula for the derivative of det(\(\lambda\) I-A). The inequality is applied to the case where A is irreducible with r and all entries integral: if the Perron-Frobenius eigenvector has integral relatively prime coordinates, then each of them is at most \(r^{k-1}\).
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 4008581
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: spectral radius
zbMATH Keywords: Perron-Frobenius eigenvector
reviewed by: Jaroslav Zemánek
MaRDI profile type: Publication
full work available at URL: 0024-3795(87)90081-4
OpenAlex ID: W2041626878
cites work: Algorithms for sliding block codes - An application of symbolic dynamics to information theory
cites work: Factors and extensions of full shifts
description: scientific article; zbMATH DE number 4008581