Linear operators that preserve pairs of matrices which satisfy extreme rank properties -- a supplementary version.
type: Item
instance of: scholarly article
title: Linear operators that preserve pairs of matrices which satisfy extreme rank properties -- a supplementary version.
zbMATH Open document ID: 1048.15005
DOI: 10.1016/S0024-3795(03)00658-X
author: Xian Zhang
published in: Linear Algebra and its Applications
publication date: 2003-12-04T00:00:00Z
review text: A pair of \(m\times n\) matrices \((A,B)\) is said to be rank-sum-maximal if \(p(A+ B)= p(A)+ p(B)\), and rank-sum-minimal if \(p(A+ B)=| p(A)- p(B)|\). The author characterizes the linear operators preserving the set of rank-sum-maximal pairs over any field and the linear operators preserving the set of rank-sum-maximal pairs are characterized by using a result about rank preservers proposed by \textit{C.-K. Li} and \textit{S. Pierce} [Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)], and thereby the linear preservers of the set of rank-sum-minimal pairs are characterized. The paper can be viewed as a supplementary version of several related results. Suppose \(F\) is an arbitrary field. Let \(M_{m,n}(F)\) be the set of all \(m\times n\) matrices over \(F\). Let \(\theta_+\) (respectively, \(\theta_-\)) be the subset of \(M_{m,n}(F)\times M_{m,n}(F)\) consisting of all rank-sum-maximal (respectively, rank-sum-minimal) pairs. Section 2 characterizes the linear preservers of the set \(\theta_+\) over any field \(F\). Section 3 characterizes the linear preservers of the set \(\theta_-\) over an field \(F\) except for \(\{0\}\).
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 2013059
zbMATH Keywords: field
zbMATH Keywords: rank-sum-maximal matrix pair
zbMATH Keywords: rank-sum-minimal matrix pair
zbMATH Keywords: linear operator
zbMATH Keywords: rank preservers
zbMATH Keywords: linear preservers
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
cites work: Linear operators that preserve pairs of matrices which satisfy extreme rank properties
cites work: Sur une généralisation du groupe orthogonal à quatre variables
cites work: Linear preservers for matrix inequalities and partial orderings
cites work: Linear Preserver Problems
cites work: Q4433359
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description: scientific article; zbMATH DE number 2013059