Index of Hadamard multiplication by positive matrices
type: Item
instance of: scholarly article
title: Index of Hadamard multiplication by positive matrices
zbMATH Open document ID: 0937.15016
DOI: 10.1016/S0024-3795(98)10215-X
author: Demetrio Stojanoff
published in: Linear Algebra and its Applications
publication date: 2000-02-22T00:00:00Z
review text: The author introduces two notions of index of a positive complex \(n\times n\) matrix \(A\) in terms of its action on the positive matrices by Hadamard multiplication: \[ \begin{aligned} I_A& =\max\{\lambda \geq 0: \|A\circ B\|\geq \lambda\|B\|, \;0 \geq B \in M_n\}, \\ II_A& =\max\{\lambda \geq 0: A\circ B \geq \lambda B, \;0 \geq B \in M_n\}.\end{aligned} \] He gives formulae to compute these indices. Originally the numbers \(I_A\) and \(II_A\), for positive matrices were studied in order to get sharp constants for some operator inequalities [see for example \textit{L. Livshits} and \textit{S. C. Ong}, ibid. 183, 117-129 (1993; Zbl 0809.47019)]. Let \(M_n\) be the \({\mathbb C}^{*}\)-algebra of \(n\times n\) complex matrices over \({\mathbb C}\), \(P(n) = \{A \in M_n: A \geq 0\}\) the set of positive (or non-negative) matrices and the map \(\Phi_{A}: M_n \to M_n\) is given by \(\Phi_{A}(B) = A\circ B\) for all \(B \in M_n\). On studying the formulae for \(I_A\) and \(II_A\) the author finds that they are given in terms of positive matrix completions and factorizations of \(A\) (see Theorem 3.11), very similar as those formulae for the norm of \(\Phi_{A}\). Furthermore, his formulae are closely related with the harmonic mean \(M:({\mathbb R}^{+})^{n} \to {\mathbb R}^{+}\) given by \[ M(d_1, \ldots d_n) = \left(\sum_{i=1}^n d_i^{-1}\right)^{-1}, \quad (d_1, \ldots d_n) \in ({\mathbb R}^{+})^{n}. \] Actually he uses its mean properties in order to compute both indexes for some examples of positive matrices (namely, diagonal or block diagonal matrices) and also to characterize those matrices \(A \in P(n)\) such that \(I_A = 0\). Moreover a complete description is given in the \(2 \times 2\) case.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1331368
zbMATH Keywords: positive matrix
zbMATH Keywords: Hadamard product
zbMATH Keywords: index
zbMATH Keywords: operator inequalities
zbMATH Keywords: matrix completions
zbMATH Keywords: factorization
zbMATH Keywords: harmonic mean
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
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full work available at URL: s0024-3795(98)10215-x
OpenAlex ID: W4206080123
description: scientific article; zbMATH DE number 1331368