Some results on matrix means
type: Item
instance of: scholarly article
title: Some results on matrix means
zbMATH Open document ID: 1450.15020
author: Mitsuru Uchiyama
published in: Advances in Operator Theory
publication date: 2020-07-27T00:00:00Z
review text: Let \(A\) and \(B\) be positive definite matrices with \(B\leq A\) where all eigenvalues lie in the range \([m,M]\) with \(0 < m < M\). It is shown that \(4mM(m+M)^{-2}B^{2}\leq mMB((m+M)I-B)^{-1}\leq mMA((m+M)I-A)^{-1}\leq A^{2}.\) This refines an inequality in [\textit{M. Fujii} et al., Nihonkai Math. J. 8, No. 2, 117--122 (1997; Zbl 0997.47505)]. The author also proves that the geometric mean \(A\#B:=A^{1/2}\left( A^{-1/2}BA^{-1/2}\right) ^{1/2}A^{1/2}\) satisfies \(A(A+B)^{\dag}B=(A\#B)(A+B)^{\dag}(A\#B),\) where \(\dag\) denotes the Moore-Penrose generalized inverse.
reviewed by: John D. Dixon
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A10
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A64
zbMATH DE Number: 7225051
zbMATH Keywords: matrix harmonic mean
zbMATH Keywords: matrix geometric mean
zbMATH Keywords: operator monotone function
MaRDI profile type: Publication
full work available at URL: s43036-019-00036-7
OpenAlex ID: W3001323611
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DOI: 10.1007/S43036-019-00036-7
description: scientific article; zbMATH DE number 7225051