The Golden-Thompson trace inequality is complemented

Fumio Hiai, Dénes Petz

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71 Citations (Scopus)

Abstract

We prove a class of trace inequalities which complements the Golden-Thompson inequality. For example, Tr(epA#epB) 2 p≤Tr eA+B holds for all p>0 when A and B are Hermitian matrices and # denotes the geometric mean. We also prove related trace inequalities involving the logarithmic function; namely p-1TrX log Y p> 2XpY p 2≤Tr X(log X+log Y)≤p-1TrX log X p 2YpX p 2 for all p>0 when X and Y are nonnegative matrices. These inequalities supply lower and upper bounds on the relative entropy.

Original languageEnglish
Pages (from-to)153-185
Number of pages33
JournalLinear Algebra and Its Applications
Volume181
Issue numberC
DOIs
Publication statusPublished - Mar 1 1993

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ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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