Positivity and conditional positivity of Loewner matrices
type: Item
instance of: scholarly article
title: Positivity and conditional positivity of Loewner matrices
zbMATH Open document ID: 1204.15039
author: Rajendra Bhatia
author: Takashi Sano
published in: Positivity
publication date: 2010-10-08T00:00:00Z
review text: Using the following well-known facts (a) the Cauchy matrix \(C = \left[\frac{1}{p_i} + \frac{1}{p_j}\right]\) is psd (positive semidefinite) for any \(n\) positive numbers \(p_1 ,\dots,p_n\), and (b) the Hadamard (entrywise) product of two psd matrices is also psd, in this paper it is elementarily shown that the Loewner matrices \(L = \left[\frac{f(p_i) - f(p_j)}{p_i - p_j} \right]\) for \(f(t) = t^r\), \(t \in (0,\infty)\), are (1) psd, (2) conditionally negative definite (cnd), and (3) conditionally positive definite (cpd) for \(r\) in \([0,1]\), \([1,2]\), and \([2,3]\), respectively. Next, for \(f\) a real valued function on any open interval, the following assertions are examined: (i) \(L\) are psd if \(f\) is operator monotone, (ii) \(L\) are cnd if \(f\) is operator convex, and (iii) \(L\) are cpd if \(f(t) = tg(t)\) for an operator convex function \(g\). It is concluded that (i) and (iii) are true, and (ii) is not true.
reviewed by: Mihail Voicu
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 42A82
zbMATH DE Number: 5797571
zbMATH Keywords: Loewner matrix
zbMATH Keywords: operator monotone
zbMATH Keywords: operator convex
zbMATH Keywords: positive semidefinite
zbMATH Keywords: conditionally positive definite
zbMATH Keywords: conditionally negative definite
zbMATH Keywords: Cauchy matrix
zbMATH Keywords: Hadamard product
MaRDI profile type: Publication
full work available at URL: s11117-009-0027-2
OpenAlex ID: W1995537216
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DOI: 10.1007/S11117-009-0027-2
description: scientific article; zbMATH DE number 5797571