Exposed faces and duality for symmetric and unitarily invariant norms
type: Item
instance of: scholarly article
title: Exposed faces and duality for symmetric and unitarily invariant norms
zbMATH Open document ID: 0808.15015
DOI: 10.1016/0024-3795(94)90499-5
author: E. Marques de Sá
published in: Linear Algebra and its Applications
publication date: 1995-03-13T00:00:00Z
review text: In this series of three papers the author investigates the faces of the unit ball of a symmetric gauge function on \(\mathbb{R}^ n\) and the unit ball of a unitarily invariant norm on the \(n \times m\) matrices. (A symmetric gauge function is also known as a norm invariant under signed permutations of the coordinates.) To study faces of the unit ball of a symmetric gauge function it is essential to study standard faces, i.e. those faces whose relative interior meets the standard cone \(D_ +\) of \(\mathbb{R}^ n\), where \(D_ +\) is the cone consisting of the nonnegative vectors with a decreasing set of coordinates. The trace of a face is the intersection of the face and the standard cone \(D_ +\). The first paper (pp. 349-395; see Zbl 0808.15013) describes the set of symmetries of a face of a symmetric gauge function in terms of its barycenter. Then it characterizes completely standard faces and the subsets of the unit ball that are traces of faces of the unit ball of a symmetric gauge function. In the second paper (pp. 451-493; see Zbl 0808.15014) the author investigates the faces of the unit ball of a unitarily invariant norm \(\psi\) on the \(n \times m\) matrices. There is a closely related symmetric gauge function \(g\) on \(\mathbb{R}^ n\) associated with \(\psi\), namely the one that satisfies \(\psi (A) = g(s(A))\) where \(s(A)\) is the vector of singular values in decreasing order. This paper gives a complete characterization of the faces for the unit ball for \(\psi\) in terms of the faces of the unit ball for \(g\). The face generated by a single matrix is characterized. The third paper (pp. 429-450) investigates the relation between the exposed faces of the unit ball for a unitarily invariant norm \(\psi\) on the \(n \times m\) matrices and those for the corresponding dual norm. The subdifferential of \(\psi\) is completely determined by the subdifferential of \(g\).
reviewed by: Tage Bai Andersen
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 52A20
zbMATH DE Number: 553971
zbMATH Keywords: symmetric gauge function
zbMATH Keywords: unitarily invariant norm
zbMATH Keywords: norm invariant
zbMATH Keywords: standard faces
zbMATH Keywords: cone
zbMATH Keywords: set of symmetries
zbMATH Keywords: singular values
zbMATH Keywords: characterization
zbMATH Keywords: exposed faces
zbMATH Keywords: subdifferential
MaRDI profile type: Publication
full work available at URL: 0024-3795(94)90499-5
OpenAlex ID: W2076534448
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description: scientific article; zbMATH DE number 553971