Geometry and the norms of Hadamard multipliers
type: Item
instance of: scholarly article
title: Geometry and the norms of Hadamard multipliers
zbMATH Open document ID: 0821.15018
DOI: 10.1016/0024-3795(93)00186-4
author: Carl C. Cowen
author: Kelly E. Debro
author: Peter D. Sepanski
published in: Linear Algebra and its Applications
publication date: 1995-05-11T00:00:00Z
review text: As a matter of fact, the norm of a matrix \(B\) as a Hadamard multiplier is the norm of the map \(X \to X \cdot B\) where ``\(\cdot\)'' is the Hadamard or entrywise product of matrices. While \textit{U. Haagerup} [Decompositions of completely bounded maps on operator algebras (Preprint)] showed that if \(B\) is an \(n \times n\) matrix, then the norm of \(B\) as a Hadamard multiplier is \(K_ B = \min \{C(S) C(R) : S^* R = B,S,R \in {\mathcal M}_ n\}\), where \(C(P) = \max \{\| P_ 1 \|_ 2, \| P_ 2 \|_ 2, \dots, \| P_ n \|_ 2\}\), the authors give a geometric interpretation to a criterion of Haagerup: If \(B\) is an \(n \times n\) matrix, then \(K_ B\) is the radius of the smallest cube that contains an ellipsoid containing each column of the matrix \(B\), and then use this interpretation to find an explicit formula for the Hadamard multiplier norms of real \(2 \times 2\) matrices. For Hermitian matrices with one positive eigenvalue, necessary and sufficient conditions are given for the norm to be the magnitude of the largest entry.
reviewed by: Yang Yingchen
Mathematics Subject Classification ID: 15A60
zbMATH DE Number: 753872
zbMATH Keywords: Hadamard product
zbMATH Keywords: factorization
zbMATH Keywords: matrix norm
zbMATH Keywords: Hadamard multiplier
zbMATH Keywords: geometric interpretation
MaRDI profile type: Publication
full work available at URL: 0024-3795(93)00186-4
OpenAlex ID: W2061168688
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Wikidata QID: Q126622076
description: scientific article; zbMATH DE number 753872