Multiplicative principal-minor inequalities for totally nonnegative matrices
type: Item
instance of: scholarly article
title: Multiplicative principal-minor inequalities for totally nonnegative matrices
zbMATH Open document ID: 1030.15021
DOI: 10.1016/S0196-8858(02)00506-7
author: Shaun M. Fallat
author: Michael Gekhtman
author: Charles R. Johnson
published in: Advances in Applied Mathematics
publication date: 2003-09-15T00:00:00Z
review text: An \(n\times n\) matrix \(A\) is called totally nonnegative (TN) if every minor of \(A\) is nonnegative. Several inequalities are known among products of principal minors, especially for positive definite and TN matrices. Examples of these inequalities are the Hadamard, Fischer and Koteljanskii inequalities. The paper under review is motivated by the question of finding the set of possible inequalities involving determinants (i.e. inequalities among principal minors) for a given class of matrices, and in particular for the class of TN matrices. The authors succeed in identifying all possible inequalities among products of principal minors for invertible TN matrices (and thus, for TN matrices by a density argument) when \(n\leq 5\), and they characterize several cases for general \(n\). In all of the cases studied by the authors the constant in the inequality is always one, and the authors conjecture that this is true in general. The characterization of the inequalities is done in terms of combinatorial objects.
reviewed by: Martín Argerami
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1981406
zbMATH Keywords: totally nonnegative matrices
zbMATH Keywords: determinant
zbMATH Keywords: principal minor
zbMATH Keywords: bidiagonal factorization
zbMATH Keywords: determinantal inequalities
zbMATH Keywords: Hadamard inequality
zbMATH Keywords: Fischer inequality
zbMATH Keywords: Koteljanskii inequalities
describes a project that uses: cdd
MaRDI profile type: Publication
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full work available at URL: s0196-8858(02)00506-7
OpenAlex ID: W2021000796
description: scientific article; zbMATH DE number 1981406