A note on the resolvent of a nonnegative matrix and its applications
type: Item
instance of: scholarly article
title: A note on the resolvent of a nonnegative matrix and its applications
zbMATH Open document ID: 1192.15015
author: Siddhartha Sahi
published in: Linear Algebra and its Applications
publication date: 2010-04-06T00:00:00Z
review text: The main result of this paper is the following. Let \(X' = (x'_{ij} )\) be a nonnegative matrix obtained from \(X = (x_{ij} )\) by decreasing a single entry \(x_{hk}\). Then for all indices \(i,j\) and for all \(t\) in \([0,1/\max (r_X ,r_{X'} ))\), where \(r_X ,r_{X'}\) are the spectral radii of \(X\) and respectively \(X'\), the entries of \(Y = (I - tX)^{ - 1}\) and \(Y' = (I - tX')^{ - 1}\) satisfy \(y_{ij} y'_{ik} \leqslant y'_{ij} y_{ik}\) and \(y_{ij} y'_{hj} \leqslant y'_{ij} y_{hj}\). From this result, two corollaries for Google page-rank model and the open Leontief model are derived.
reviewed by: Mihail Voicu
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B36
Mathematics Subject Classification ID: 91A40
Mathematics Subject Classification ID: 91B66
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 5689419
zbMATH Keywords: nonnegative matrix
zbMATH Keywords: inverse \(M\)-matrix
zbMATH Keywords: Leontief model
zbMATH Keywords: Google page-rank
MaRDI profile type: Publication
full work available at URL: j.laa.2009.11.004
OpenAlex ID: W2076962931
cites work: A strategic market game with complete markets
cites work: Nonnegative Square Matrices
cites work: Competing for customers in a social network
cites work: A theory of money and financial institutions. 28. The noncooperative equilibria of a closed trading economy with market supply and bidding strategles
cites work: Note: Some Conditions of Macroeconomic Stability
cites work: Nonnegative Matrices whose Inverses are M-Matrices
cites work: Adjustment of an Inverse Matrix Corresponding to a Change in One Element of a Given Matrix
cites work: Logarithmic Convexity of Perron-Frobenius Eigenvectors of Positive Matrices
cites work: The non-cooperative equilibria of a trading economy with complete markets and consistent prices
cites work: The weak law of large numbers for nonnegative summands
DOI: 10.1016/J.LAA.2009.11.004
description: scientific article; zbMATH DE number 5689419