New bounds for Perron root of a nonnegative matrix
type: Item
instance of: scholarly article
title: New bounds for Perron root of a nonnegative matrix
zbMATH Open document ID: 1127.15014
DOI: 10.1007/BF02831980
author: Weiguo Li
author: Jinhai Chen
published in: Journal of Applied Mathematics and Computing
publication date: 2007-04-05T00:00:00Z
review text: Let \(A\) be a nonnegative \(n\times n\) matrix, \(r\) be the Perron root of \(A\), which is the real eigenvalue of \(A\) of largest in absolute values of all eigenvalues of \(A\). It is shown in this paper that if \(f\) is a polynomial such that \(f(r)\neq0\) and the matrix \(B=f(A)\) is nonnegative, all row sums are nonzero, then for integers \(k\geq0\) and \(m\geq1\) \[ \min_i(\frac{r_i(A^mB^k)}{r_i(B^k)})^{1/m}\leq r\leq\max_i(\frac{r_i(A^mB^k)}{r_i(B^k)})^{1/m}, \] where \(r_i\) denotes the row sum of the \(i\)th row of the matrix so indicated. The inequalities remain valid if the row sums \(r_i\) are replaced by the column sums \(c_i\). Known bounds for the Perron root by \textit{G. Frobenius} [Berl. Ber. 1912, 456--477 (1912; JFM 43.0204.09)] and others may be derived by choosing \(m=1\) and \(B=A^2\).
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 5140499
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: bounds for Perron root
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02831980
OpenAlex ID: W2056773421
description: scientific article; zbMATH DE number 5140499