Riemannian geometry and matrix geometric means
type: Item
instance of: scholarly article
title: Riemannian geometry and matrix geometric means
zbMATH Open document ID: 1088.15022
author: Rajendra Bhatia
author: John Holbrook
published in: Linear Algebra and its Applications
publication date: 2006-03-22T00:00:00Z
review text: The goals of this article are partly expository but include an analysis of recent approaches to the definition of a geometric mean for three (or more) positive definite matrices. The authors first review some standard constructions from Riemannian geometry. Then they show how these constructions lead to a better and deep understanding of the geometric mean of two positive definite matrices. This allows them to treat the problem of extending these ideas to three matrices. The authors notice that the several definitions of geometric means appeared in the literature do not readily extend to three matrices. In some recent papers the geometric mean of \(A\) and \(B\) is explained as the midpoint of the geodesic (with respect to a natural Riemannian metric) joining \(A\) and \(B\). This new understanding of the geometric mean suggests some natural definitions for a geometric mean of three positive definite matrices.
reviewed by: Fozi Dannan
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 53B21
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 5014301
zbMATH Keywords: positive definite matrix
zbMATH Keywords: geometric mean
zbMATH Keywords: Riemannian manifold
zbMATH Keywords: semi-parallelogram law
Wikidata QID: Q115344936
MaRDI profile type: Publication
full work available at URL: j.laa.2005.08.025
OpenAlex ID: W1974200742
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DOI: 10.1016/J.LAA.2005.08.025
description: scientific article; zbMATH DE number 5014301