Geometric means
type: Item
instance of: scholarly article
title: Geometric means
zbMATH Open document ID: 1063.47013
author: Chi-Kwong Li
author: Roy Mathias
author: T. Ando
published in: Linear Algebra and its Applications
publication date: 2004-08-06T00:00:00Z
review text: Let \(G(A,B)\) be the geometric mean of two \(n\times n\) positive semidefinite matrices \(A\) and \(B\). The authors extend the definition of \(G\) to any number of \(n\times n\) positive semidefinite matrices inductively. Suppose that for some \(k\geq 2\), the geometric mean \(G(A_1,A_2,\dots,A_k)\) of any \(k\) positive semidefinite matrices \(A_1,A_2,\dots,A_k\) has been defined. Let \(A=(A_1,A_2,\dots,A_k,A_{k+1})\) be a \((k+1)\)-tuple of \(n\times n\) positive semidefinite matrices. Define \(T(A)\equiv (G((A_i)_{i\neq 1}), G((A_i)_{i\neq 2}),\dots,G((A_i)_{i\neq k+1}))\). The authors show that the sequence \((T^r(A))_{r=1}^{\infty}\) has a limit of the form \((\tilde A,\dots,\tilde A)\) and define \(G(A_1,A_2,\dots,A_k,A_{k+1})=\tilde A\). The definition given here is the only one in the literature that has the properties that one would expect from a geometric mean. The authors also prove some new properties of the geometric mean of two matrices, and give some simple computational formulae related to them for \(2\times 2\) matrices.
Mathematics Subject Classification ID: 47A64
Mathematics Subject Classification ID: 47A63
Mathematics Subject Classification ID: 15A45
zbMATH DE Number: 2083518
zbMATH Keywords: positive semidefinite matrix
zbMATH Keywords: geometric mean
zbMATH Keywords: matrix square root
zbMATH Keywords: matrix inequality
zbMATH Keywords: spectral radius
reviewed by: Yiu-Tung Poon
describes a project that uses: mctoolbox
MaRDI profile type: Publication
full work available at URL: j.laa.2003.11.019
OpenAlex ID: W4206551434
cites work: Series and parallel addition of matrices
cites work: Symmetric function means of positive operators
cites work: Q4748635
cites work: Calculation of Gauss Quadrature Rules
cites work: Q4864704
cites work: Q3699471
cites work: Extremal Characterizations of the Schur Complement and Resulting Inequalities
cites work: Hermitian semidefinite matrix means and related matrix inequalities—an introduction
DOI: 10.1016/J.LAA.2003.11.019
description: scientific article; zbMATH DE number 2083518