Qualitative convergence of matrices
type: Item
instance of: scholarly article
title: Qualitative convergence of matrices
zbMATH Open document ID: 1176.15012
author: Olga Pryporova
published in: Linear Algebra and its Applications
publication date: 2009-06-04T00:00:00Z
review text: A complex square matrix \(A\) is stable if all its eigenvalues lie in the open left half-plane of \(\mathbb{C}\), and convergent if they lie within the open unit disk. Qualitative analysis, where the entries of \(A\) are not specified, arose from the study of stability problems in economics and ecology. In the continuous time case, only the signs of the entries of the real matrix \(A\) (its sign pattern) are given. In the discrete time case, what is given for each entry of \(A\) is whether its modulus is \(0\) or \(1\) or in \((0,1)\) or greater than \(1\) (the modulus pattern of \(A\)). Such qualitative information about \(A\) can be used to determine the stability or convergence of \(A\). In this paper the author introduces several types of potential convergence, the basic definition of which is: a modulus pattern \(Z\) is potentially convergent if there exists a convergent matrix with the modulus pattern \(Z\). All of these different types are characterized for matrices of size 2. Potential absolute convergence is shown to be equivalent to potential diagonal convergence and their complete characterization for size \(n\) is given.
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 05C50
Mathematics Subject Classification ID: 15B35
zbMATH DE Number: 5561607
zbMATH Keywords: modulus convergence
zbMATH Keywords: modulus pattern
zbMATH Keywords: potential convergence
zbMATH Keywords: potential absolute convergence
zbMATH Keywords: \(D\)-convergence
zbMATH Keywords: diagonal convergence
zbMATH Keywords: qualitative convergence
zbMATH Keywords: qualitative stability
zbMATH Keywords: eigenvalues
reviewed by: Rabe von Randow
MaRDI profile type: Publication
full work available at URL: j.laa.2009.02.003
OpenAlex ID: W2017367519
cites work: Q3943082
cites work: Positive feedback may sometimes promote stability
cites work: Q4679023
cites work: On the potential stability of star sign pattern matrices
cites work: Matrix Analysis
cites work: Some Sign Patterns that Preclude Matrix Stability
cites work: When is a Matrix Sign Stable?
cites work: Nested sequences of principal minors and potential stability
cites work: The potentially stable tree sign patterns for dimensions less than five
cites work: Qualitative stability of discrete-time systems
cites work: Q4937699
cites work: Q3972038
cites work: Stronger-than-Lyapunov notions of matrix stability, or how ``flowers'' help solve problems in mathematical ecology
cites work: Q3490133
cites work: On another approach to the Schur-Cohn criterion
DOI: 10.1016/J.LAA.2009.02.003
Recommended article: Q3490133
Recommended article: Q3713126
Recommended article: Q5312803
Recommended article: Convergence of matrix powers
Recommended article: Convergence of splittings of matrices
Recommended article: Some Convergence Properties of Matrix Sets
Recommended article: Almost convergence and some matrix transformations
Recommended article: Convergence of a class of Toeplitz type matrices
description: scientific article; zbMATH DE number 5561607