Some results on matrix monotone functions
type: Item
instance of: scholarly article
title: Some results on matrix monotone functions
zbMATH Open document ID: 0679.15026
DOI: 10.1016/0024-3795(89)90577-6
author: Man Kam Kwong
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: A realvalued function \(f\) defined on the nonnegative reals is called matrix monotone if for all \(n\) and all Hermitian \(n\times n\) matrices with \(0\leq B\leq A\) then \(f(B)\leq f(A)\). The matrix monotone functions were characterized by \textit{C. Löwner} [Math. Z. 38, 177-216 (1934; Zbl 0008.11301)] via an integral representation. Using this the author gives alternative proofs of some of the known results in this area, including the arithmetic mean/harmonic mean inequality, \textit{F. Hansen}'s inequality [Math. Ann. 246, 249-250 (1984; Zbl 0407.47012)] and some of the authors own results, and he extends some of these. The author extends Lyapunov's classical theorem and shows: Let \(A\) and \(P\) be positive definite. If \(f\) or \(1/f\) is matrix monotone then the solution \(X\) of \(AX+XA=f(A)P+Pf(A)\) is positive definite.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 26D15
zbMATH DE Number: 4112793
zbMATH Keywords: matrix monotone functions
zbMATH Keywords: arithmetic mean/harmonic mean inequality
reviewed by: Tage Bai Andersen
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90577-6
OpenAlex ID: W2093461426
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description: scientific article; zbMATH DE number 4112793