Linear/additive preservers of ranks 2 and 4 on alternate matrix spaces over fields
type: Item
instance of: scholarly article
title: Linear/additive preservers of ranks 2 and 4 on alternate matrix spaces over fields
zbMATH Open document ID: 1059.15010
author: Xian Zhang
published in: Linear Algebra and its Applications
publication date: 2004-11-26T00:00:00Z
review text: A square matrix \(A\) over a field \(F\) is ``alternate'' if it is skew-symmetric and all diagonal entries are \(0\) (the latter condition is implied by the former condition except when \text{char}\((F)=2\)). The \(F\)-space of all \(n\times n\) alternate matrices over \(F\) is denoted by \(K_{n}(F)\). When \(n=4\) we define \(q\) to be the linear transformation \(q\) of \(K_{4}(F)\) which interchanges the \((1,4)\)th and \((4,1)\)th entries of \(A\) with the \((2,3)\)th and \((3,2)\)th entries, respectively. The main results of this paper are the following. If \(f\) is a linear operator on \(K_{n}(F)\) with \(n\geq4\), then \(f\) preserves the ranks of matrices of ranks \(2\) or \(4\) if and only if there exists \(\gamma\in F\) and a nonsingular \(S\in M_{n}(F)\) such that either (i) \(f(A)=\gamma SAS^{T}\) for all \(A\); or (ii) \(n=4\) and \(f(A)=\gamma Sq(A)S^{T}\) for all \(A\). Moreover, provided \(F\) is not isomorphic to one of its own proper subfields, a similar conclusion holds under the weaker assumption that \(f\) is only an additive map. In this case \(f\) preserves the ranks of matrices of ranks \(2\) and \(4\) if and only if \(f\) is a composite of a field automorphism followed by a function of type (i) or (ii).
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A03
zbMATH DE Number: 2119125
zbMATH Keywords: skew-symmetric matrix
zbMATH Keywords: linear preserver
zbMATH Keywords: rank
zbMATH Keywords: additive preserver
zbMATH Keywords: alternate matrix
zbMATH Keywords: linear transformation
MaRDI profile type: Publication
full work available at URL: j.laa.2004.06.001
OpenAlex ID: W1969304006
cites work: Rank preservers on spaces of symmetric matrices
cites work: Linear preservers on matrices
cites work: Some general techniques on linear preserver problems
cites work: Automorphisms of the Field of Complex Numbers
cites work: Linear Preserver Problems
cites work: Linear maps on skew symmetric matrices: The invariance of elementary symmetric functions
cites work: Additive mappings preserving operators of rank one
cites work: On Hua's fundamental theorem of the geometry of rectangular matrices
cites work: Q4885576
cites work: Linear transformations on Grassmann spaces
cites work: Linear Transformations on Grassmann Spaces III
cites work: Additive preservers of rank-additivity on the spaces of symmetric and alternate matrices.
cites work: Linear operators that preserve pairs of matrices which satisfy extreme rank properties -- a supplementary version.
DOI: 10.1016/J.LAA.2004.06.001
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description: scientific article; zbMATH DE number 2119125