On maximum principles for monotone matrices
type: Item
instance of: scholarly article
title: On maximum principles for monotone matrices
zbMATH Open document ID: 0587.15014
DOI: 10.1016/0024-3795(86)90021-2
author: Gisbert Stoyan
published in: Linear Algebra and its Applications
publication date: 1986-01-01T00:00:00Z
review text: The (n,n)-matrix A satisfies the maximum principle if \(Ay=f\) and \(f\geq 0\) implies \(y\geq 0\), and moreover \(\max \{y_ i|\) \(i\in N\}=\max \{y_ i| i\in N^+(f)\}\) with \(N=\{1,...,n\}\) and \(N^+(f)=\{j\in N| f_ j>0\}\). Define \(A^{(+)}=a_{ij}\) for \(i\neq j\) and \(a_{ij}>0\) and 0 elsewhere. The following holds. Let A be nonsingular with nonnegative inverse and such that \(A-A^{(+)}\) is either nonsingular, or singular and irreducible. Moreover, let \(A-A^{(+)}\) have nonnegative row sums. Then A satisfies the maximum principle. Analogous properties of the solutions to linear equations with an irreducible M-matrix are studied in connection with the open Leontief input-output model [the reviewer, ibid. 26, 175-201 (1979; Zbl 0409.90027)].
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 93D25
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 3941716
zbMATH Keywords: monotone matrix
zbMATH Keywords: Drazin inverse
zbMATH Keywords: maximum principle
zbMATH Keywords: nonnegative inverse
zbMATH Keywords: irreducible
zbMATH Keywords: M-matrix
zbMATH Keywords: Leontief input-output model
MaRDI profile type: Publication
full work available at URL: 0024-3795(86)90021-2
OpenAlex ID: W2034843240
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description: scientific article; zbMATH DE number 3941716