A representation theorem for (tr \(A^ p)^{1/p}\)
type: Item
instance of: scholarly article
title: A representation theorem for (tr \(A^ p)^{1/p}\)
zbMATH Open document ID: 0632.15011
DOI: 10.1016/0024-3795(87)90030-9
author: Jan R. Magnus
published in: Linear Algebra and its Applications
publication date: 1987-01-01T00:00:00Z
review text: Let A and B be \(n\times n\) positive semidefinite real matrices, \(p>1\), and let \(q=p/(p-1)\). The main result of the paper is the following quasilinear representation of (tr \(A^ p)^{1/p}:\) \(tr(A^ p)^{1/p}=_{tr B^ q=1}tr(AB).\) Using this it is shown that \(tr(AB)\leq (tr A^ p)^{1/p}(tr B^ q)^{1/q}\) and \((tr(A+B)^ p)^{1/p}\leq (tr A^ p)^{1/p}+(tr B^ p)^{1/p}\) which are matrix analogues of the Hölder and Minkowski inequalities, respectively.
Mathematics Subject Classification ID: 15A45
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 4027682
zbMATH Keywords: trace
zbMATH Keywords: Hölder inequality
zbMATH Keywords: Minkowski inequality
zbMATH Keywords: positive semidefinite
zbMATH Keywords: quasilinear representation
reviewed by: Ali-Akbar Jafarian
MaRDI profile type: Publication
OpenAlex ID: W2113383060
cites work: Q5606006
cites work: Q5812325
cites work: Inequalities: theory of majorization and its applications
description: scientific article; zbMATH DE number 4027682