Multiplicative preservers on semigroups of matrices
type: Item
instance of: scholarly article
title: Multiplicative preservers on semigroups of matrices
zbMATH Open document ID: 1025.15001
DOI: 10.1016/S0024-3795(02)00346-4
author: Wai-Shun Cheung
author: Chi-Kwong Li
author: Shaun M. Fallat
published in: Linear Algebra and its Applications
publication date: 2003-02-05T00:00:00Z
review text: A ``linear preserver'' is a linear function of the space of all matrices (of given dimension over a given field) into itself which maps a prescribed subset \(\mathcal{P}\) into itself (for example, \(\mathcal{P}\) might be the set of all nonsingular matrices). Linear preservers have been widely studied and a typical theme is that for many interesting choices of \(\mathcal{P}\) they are necessarily of one of the forms: \(A\mapsto PAQ\) or \(A\mapsto PA^{T}Q\) for invertible \(P\) and \(Q\) [see, for example, \textit{C. K. Li} and \textit{S. Pierce}, Am. Math. Mon. 108, 591-605 (2001; Zbl 0991.15001)]. In the present paper the authors consider a related problem of ``multiplicative preservers''. Let \(\mathbf{S}\) be either a field or the semigroup of nonnegative reals or nonnegative rationals, and let \(\mathcal{R}\) be a subsemigroup of the semigroup \(M_{n}(\mathbf{S})\) of all \(n\times n\) matrices over \({\mathbf{S}}\). Consider a multiplicative map \(\phi\) of \(\mathcal{R}\) into itself. The authors prove that for certain subsets \(\mathcal{P\subseteq R}\), if \(\phi\) preserves \(\mathcal{P}\), then one of the following typically holds: (i) there exists an invertible \(S\) and a nonzero ``additive-multiplicative'' map \(f:\mathbf{S}\rightarrow\mathbf{S}\) such that \(\phi\) has one of the forms \(A\mapsto S^{-1}[f(a_{ij})]S\) or \(A\mapsto S^{-1}[f(\det A_{ij})S\) where \(A=[a_{ij}]\) and \(A_{ij}\) is the submatrix obtained by deleting the \(i\)th row and \(j\)th column (the precise definition of ``additive-multiplicative'' is too complicated to give here); (ii) there exists an invertible \(S\), an integer \(q\) with \(0\leq q\leq n\) and a multiplicative map \(\psi:M_{n}(\mathbf{S})\rightarrow M_{n-q}(\mathbf{S})\) such that \(\phi\) has the form \(A\mapsto S^{-1}(I_{q}\oplus\psi(A))S\); (iii) \(\phi\) is identically \(0\) on all matrices of rank \(\leq p\) where \(p\) depends on the subset preserved. For example, they prove such a theorem when \(\mathcal{P}…
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1864740
zbMATH Keywords: semigroups of matrices
zbMATH Keywords: linear preserver
zbMATH Keywords: multiplicative preservers
zbMATH Keywords: rank-\(k\) matrices
zbMATH Keywords: totally nonnegative matrices
zbMATH Keywords: Hermitian matrices
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1864740