The matrix geometric mean of parameterized, weighted arithmetic and harmonic means
type: Item
instance of: scholarly article
title: The matrix geometric mean of parameterized, weighted arithmetic and harmonic means
zbMATH Open document ID: 1222.15036
author: Sejong Kim
author: Yongdo Lim
author: Jimmie D. Lawson
published in: Linear Algebra and its Applications
publication date: 2011-07-15T00:00:00Z
review text: For positive definite matrices \(C\) and \(D\), the matrix geometric mean \(C \sharp D\) is the metric midpoint of the of arithmetic mean \(A = \frac12(C + D)\) and the harmonic mean \(H = 2(C^{-1} + D^{-1})^{-1}\) for the trace metric. The authors consider the more general construction of taking the geometric mean of the weighted \(n\)-variable arithmetic and harmonic means. More precisely, for \(\omega \in (0, 1)^m\) with \(\|\omega\|_1=1\) and positive definite matrices \(A_1,\dots,A_m\) with \(A = (A_1,\dots,A_m)\) they introduce the weighted \(A\sharp H\)-mean to be the matrix geometric mean of the weighted arithmetic and harmonic means: \(\mathcal L(\omega;A) := \left( \sum_i \omega_i A_i\right) \sharp \left( \sum_i \omega_i A_i^{-1}\right)^{-1}\). Many properties of this weighted mean are presented, and it is interpreted via the Kullback-Leibler divergence from probability theory and information theory.
reviewed by: Igor Klep
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 47A64
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 5924576
zbMATH Keywords: matrix mean
zbMATH Keywords: positive definite matrix
zbMATH Keywords: weighted mean
zbMATH Keywords: Kullback-Leibler divergence
zbMATH Keywords: matrix geometric mean
zbMATH Keywords: arithmetic mean
zbMATH Keywords: harmonic mean
zbMATH Keywords: trace metric
MaRDI profile type: Publication
full work available at URL: j.laa.2011.04.010
OpenAlex ID: W2031138269
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DOI: 10.1016/J.LAA.2011.04.010
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description: scientific article; zbMATH DE number 5924576