Cross-positive matrices revisited
type: Item
instance of: scholarly article
title: Cross-positive matrices revisited
zbMATH Open document ID: 0828.15018
DOI: 10.1016/0024-3795(93)00364-6
author: Peter Gritzmann
author: Victor Klee
author: Bit-Shun Tam
published in: Linear Algebra and its Applications
publication date: 1995-09-06T00:00:00Z
review text: Let \(\Pi(K)\) for a closed, pointed \(n\)-dimensional convex cone \(K\) in \(\mathbb{R}^n\) denote the set of all \(n\times n\) matrices mapping \(K\) into itself, \(\Sigma(K)\) the set of cross-positive matrices, and \(\Lambda\) the set of real multiples of the identity matrix. The authors answer a question of \textit{H. Schneider} and \textit{M. Vidyasagar} [SIAM J. Numer. Anal. 7, 508-519 (1970; Zbl 0245.15008)] by showing that \(\Pi(K)+ \Lambda\neq \Sigma(K)\) for almost all \(K\). They also show a weaker equality of \textit{R. Stern} and \textit{H. Wolkowicz} [SIAM J. Matrix Anal. Appl. 15, 755-778 (1994; Zbl 0806.15007)] fails for almost all \(K\).
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 778260
zbMATH Keywords: lineality space
zbMATH Keywords: convex cone
zbMATH Keywords: cross-positive matrices
reviewed by: Ki Hang Kim
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)00364-6
OpenAlex ID: W2020981363
cites work: Theory of cones
cites work: Cones, matrices and mathematical programming
cites work: Q3996218
cites work: Q3943082
cites work: Monotonie und Randspektrum bei vollstetigen Operatoren
cites work: Quasimonotonie und Ungleichungen in halbgeordneten Räumen
cites work: Die meisten konvexen Körper sind glatt, aber nicht zu glatt
cites work: Q5547252
cites work: On the automorphism group of a cone
cites work: Some new results on smoothness and rotundity in normed linear spaces
cites work: Some characterizations of convex polyhedra
cites work: On faces generated by certain positive linear transformations
cites work: Indecomposable cones
cites work: New Characterizations of Polyhedral Cones
cites work: Q5638112
cites work: Cross-Positive Matrices
cites work: Exponential Nonnegativity on the Ice Cream Cone
cites work: Trust Region Problems and Nonsymmetric Eigenvalue Perturbations
cites work: A note on cross-positive matrices
cites work: Some results on cross-positive matrices
cites work: Some results of polyhedral cones and simplicial cones
cites work: On the duality operator of a convex cone
cites work: On the distinguished eigenvalues of a cone-preserving map
cites work: On the structure of the cone of positive operators
cites work: On point classification in convex sets.
description: scientific article; zbMATH DE number 778260