The inverse problem of geometric and golden means of positive definite matrices
type: Item
instance of: scholarly article
title: The inverse problem of geometric and golden means of positive definite matrices
zbMATH Open document ID: 1115.15013
author: Yongdo Lim
author: Hosoo Lee
published in: Archiv der Mathematik
publication date: 2007-03-02T00:00:00Z
review text: Given positive definte matrices \(A\) and \(B\), the inverse problem of the geometric and golden means consists in finding positive definite matrices \(X\) and \(Y\) such that: (1) \(X*Y=A\), (2) \((X+X*(4Y-3X))/2=B\), where \(X*Y=X^1{}^/{}^2 (X{}^-{}^1{}^/{}^2 Y X{}^-{}^1{}^/{}^2){}^1{}^/{}^2 X{}^1{}^/{}^2\) defines the geometric mean of \(X\) and \(Y\). It is shown that system (1), (2) is solvable (respectively uniquely solvable) iff \(A \leqslant 3{}^1{}^/{}^2 B \leqslant 2A\) (respectively \(A \leqslant 3{}^1{}^/{}^2 B \leqslant 3{}^1{}^/{}^2 A\)).
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A29
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 26E60
zbMATH DE Number: 5130242
zbMATH Keywords: positive definite matrix
zbMATH Keywords: geometric mean
zbMATH Keywords: golden mean
zbMATH Keywords: inverse problem
zbMATH Keywords: nonlinear matrix equation
reviewed by: Mihail Voicu
MaRDI profile type: Publication
full work available at URL: s00013-006-1934-0
OpenAlex ID: W2093929111
DOI: 10.1007/S00013-006-1934-0
description: scientific article; zbMATH DE number 5130242