A note on the rank-subtractivity ordering
type: Item
instance of: scholarly article
title: A note on the rank-subtractivity ordering
zbMATH Open document ID: 0941.15011
DOI: 10.1016/S0024-3795(98)10236-7
author: Jürgen Gross
published in: Linear Algebra and its Applications
publication date: 2000-08-13T00:00:00Z
review text: Among known partial orderings of the set \(\mathbb{C}_{m,n}\) of all complex \(m\times n\) matrices are \(A{^{rs}\leq} B\) (``rank-subtractivity'') defined by \(A{^{rs}\leq} B\Leftrightarrow \text{rk}(B-A)=\text{rk}(B)-\text{rk}(A)\), and \(A{^\sigma\leq}B\) (``singular values'') defined by \(A{^\sigma\leq}B\Leftrightarrow A{^{rs}\leq}B\) and \(\sigma(A)\subseteq\sigma(B)\), where \(\sigma(X)\) is the set of singular values of \(X\). The author introduces a new partial ordering \(A{^{\sigma_1}\leq}B\) intermediate between these two: \(A{^{\sigma_1}\leq}B\Leftrightarrow A{^{rs}\leq}B\) and \(\sigma_1(A) \leq\sigma_1(B)\), where \(\sigma_1(X)\) is the largest singular value of \(X\). He investigates its formal properties and its relations to other partial orderings. An application to multivariate analysis is given.
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A09
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 62H10
zbMATH DE Number: 1331297
zbMATH Keywords: rank-subtractivity ordering
zbMATH Keywords: complex matrices
zbMATH Keywords: singular values
zbMATH Keywords: partial orderings
zbMATH Keywords: multivariate analysis
reviewed by: G. E. Wall
MaRDI profile type: Publication
cites work: Partial orderings of matrices referring to singular values or eigenvalues
cites work: Left-star and right-star partial orderings
cites work: Some properties of matrix partial orderings
cites work: Natural structures on semigroups with involution
cites work: On semi-star-related matrices
cites work: Q4209854
cites work: Partial orderings, preorderings, and the polar decomposition of matrices
cites work: Q3884160
cites work: Matrix Analysis
cites work: Inequalities: theory of majorization and its applications
cites work: Q4004320
cites work: The minus partial order and the shorted matrix
cites work: On group inverses and the sharp order
cites work: Partial orders based on outer inverses
cites work: The natural partial order on a regular semigroup
Recommended article: Note on an Ordering Theorem for Subfields
Recommended article: Ordinal notions of submodularity
Recommended article: A note on orderings on algebraic varieties
Recommended article: Ranking functions and axioms for linear orders
Recommended article: A note on submodularity preserved involving the rank functions
Recommended article: Rank-Width and Well-Quasi-Ordering
Recommended article: On some characterizations of the ''star'' partial ordering for matrices and rank subtractivity
Recommended article: On quantifier-rank equivalence between linear orders
Recommended article: Q5749335
Recommended article: A note on number of orderings of higher level
description: scientific article; zbMATH DE number 1331297