TY - JOUR

T1 - An arithmetic-geometric mean inequality for products of three matrices

AU - Israel, Arie

AU - Krahmer, Felix

AU - Ward, Rachel

N1 - Publisher Copyright:
© 2015 Published by Elsevier Inc.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

AB - 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.

KW - Arithmeticgeometric mean inequality

KW - Linear algebra

KW - Norm inequalities

UR - http://www.scopus.com/inward/record.url?scp=84942531979&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2015.09.013

DO - 10.1016/j.laa.2015.09.013

M3 - Article

AN - SCOPUS:84942531979

SN - 0024-3795

VL - 488

SP - 1

EP - 12

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -