On partial orderings on the set of rectangular matrices
type: Item
instance of: scholarly article
title: On partial orderings on the set of rectangular matrices
zbMATH Open document ID: 0833.15012
DOI: 10.1016/0024-3795(93)00212-I
author: Jan Hauke
author: Augustyn Markiewicz
published in: Linear Algebra and its Applications
publication date: 1995-05-23T00:00:00Z
review text: Let \(\mathbb{C}_{m,n}\) and \(\mathbb{C}_n\) stand for the sets of \(m \times n\) and \(n \times n\) complex matrices respectively. The subset of \(\mathbb{C}_n\) consisting of Hermitian nonnegative definite matrices will be denoted by \(\mathbb{C}^\geq_n\), and its subset consisting of positive definite matrices by \(\mathbb{C}^>_n\). The authors introduce and investigate a new partial ordering on \(\mathbb{C}_{m,n}\). They give a characterization of this ordering and show that it collapses to the Loewner ordering when applied to \(C^\geq_n\). Moreover they give the comparison with the star ordering in general, and an alternative characterization in terms of the polar decomposition of matrices. Finally in the last section they consolidate the results of this paper in a tabular comparison.
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A09
zbMATH DE Number: 757616
zbMATH Keywords: rectangular matrices
zbMATH Keywords: Hermitian nonnegative definite matrices
zbMATH Keywords: partial ordering
zbMATH Keywords: Loewner ordering
zbMATH Keywords: star ordering
zbMATH Keywords: polar decomposition of matrices
reviewed by: Takayuki Nôno
MaRDI profile type: Publication
cites work: Mean square error matrix improvements and admissibility of linear estimators
cites work: Q4061081
cites work: Natural structures on semigroups with involution
cites work: On some characterizations of the ''star'' partial ordering for matrices and rank subtractivity
cites work: Matrix Analysis
cites work: Q4040931
cites work: Inequalities: theory of majorization and its applications
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description: scientific article; zbMATH DE number 757616