An arithmetic-geometric mean inequality for products of three matrices

Arie Israel, Felix Krahmer, Rachel Ward

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

Consider the following noncommutative arithmetic-geometric mean inequality: Given positive-semidefinite matrices A1,...,An, the ≤ m≤n:1;bsupesup ;bsupesup bsup esuphellip bsupesup=1n(Equation presented)A;bsupbsub Aj2...Ajm≥(n-m)!n!∑j1,j2,...,jm=1all distinctn(Equation presented)Aj1Aj2...Ajm(Equation presented), where (Equation presented) denotes a unitarily invariant norm, including the operator norm and Schatten p-norms as special cases. While this inequality in full generality remains a conjecture, we prove that the inequality holds for products of up to three matrices, m≤3. The proofs for m=1,2 are straightforward; to derive the proof for m=3, we appeal to a variant of the classic Araki-Lieb-Thirring inequality for permutations of matrix products.

Original languageEnglish
Pages (from-to)1-12
Number of pages12
JournalLinear Algebra and Its Applications
Volume488
DOIs
StatePublished - 1 Jan 2016

Keywords

  • Arithmeticgeometric mean inequality
  • Linear algebra
  • Norm inequalities

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