A link between the Perron-Frobenius theorem and Perron's theorem for difference equations
type: Item
instance of: scholarly article
title: A link between the Perron-Frobenius theorem and Perron's theorem for difference equations
zbMATH Open document ID: 1209.15036
author: Mihály Pituk
published in: Linear Algebra and its Applications
publication date: 2011-01-07T00:00:00Z
review text: The paper deals with the growth of the solutions of the difference equation \[ x_{n+1}=A_nx_n, \;\;n \in \mathbb{N}, \] where the coefficients \(A_n\) are complex matrices, of size \(k \times k\), asymptotically constant, that is, \[ \lim_{n \rightarrow \infty} A_n = A, \] for some \(k \times k\) complex matrix \(A\). In this work, the author is interested in those solutions \(x=(x_n)_{n \in \mathbb{N}}\) of the difference equation which are nonnegative, that is, \(x_n \geq 0\) for all \(n \in \mathbb{N}\). The main result of this paper establishes that if \(x=(x_n)_{n \in \mathbb{N}}\) is a nonnegative solution of the difference equation, then either {\parindent6.5mm \begin{itemize}\item[(i)] \(x_n=0\) for all large \(n\), or \item[(ii)] the limit \(\rho(x)=\lim_{n \rightarrow \infty} \sqrt[n]{\|x_n \|}\) is an eigenvalue of \(A\) with a nonnegative eigenvector. \end{itemize}} This result provides not only new information about the nonnegative solutions from the Perron type theorem, but, moreover, it contains the weak form of the Perron-Frobenius theorem (the spectral radius of a nonnegative matrix \(A\) is an eigenvalue of \(A\) with a nonnegative eigenvector) as a special case.
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 39A06
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 5833887
zbMATH Keywords: nonnegative matrices
zbMATH Keywords: difference equations
zbMATH Keywords: spectral radius
zbMATH Keywords: Perron-Frobenius theorem
zbMATH Keywords: iterations
zbMATH Keywords: nonnegative eigenvector
reviewed by: Juan Ramón Torregrosa Sánchez
MaRDI profile type: Publication
full work available at URL: j.laa.2010.09.002
OpenAlex ID: W2030874011
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DOI: 10.1016/J.LAA.2010.09.002
description: scientific article; zbMATH DE number 5833887