The isometries and the \(G\)-invariance of certain seminorms.
type: Item
instance of: scholarly article
title: The isometries and the \(G\)-invariance of certain seminorms.
zbMATH Open document ID: 1033.15023
DOI: 10.1016/S0024-3795(03)00544-5
author: Boris Lavrič
published in: Linear Algebra and its Applications
publication date: 2003-11-19T00:00:00Z
review text: Let \({\mathbb F}\) be either \({\mathbb R}\) or \({\mathbb C}\), and let \(G\) be a subgroup of the general linear group \(\text{GL}_n({\mathbb F})\). Given an \(m\times n\) matrix \(A\) over~\({\mathbb F}\), let \(\| \cdot\| _{A,\infty} \) be the seminorm on \({\mathbb F}^n\) defined by \(\| x\| _{A,\infty}=\| Ax\| _\infty\) for all \(x\in {\mathbb F}^n\). The author characterizes the linear isometries for the seminorm \(\| \cdot\| _{A,\infty}\) and shows that the group \({\mathcal G}_{A,\infty}\) of isometries for the norm \(\| \cdot\| _{A,\infty}\) can be embedded in the group \(\text{GP}_n({\mathbb F})\) of general permutation matrices. Further, conditions on~\(A\) for which \(\| \cdot\| _{A,\infty}\) is \(G\)@-invariant are studied. As a special case the matrices \(A\) for which \(\| \cdot\| _{A,\infty}\) is absolute or a symmetric gauge function are described. The results answer an open question from \textit{R. Hemasinha} [Linear Multilinear Algebra 35, 135--151 (1993; Zbl 0790.15025)] of characterizing matrices \(A\in {\mathbb R}^{n\times m}\) for which \(\| \cdot\| _{A,\infty}\) on~\({\mathbb R}^n\) are sign invariant.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 52A21
zbMATH DE Number: 2005991
zbMATH Keywords: linear isometry
zbMATH Keywords: \(G\)-invariance
zbMATH Keywords: seminorm
reviewed by: A. R. Alimov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 2005991