A note on the non-commutative arithmetic-geometric mean inequality

2014/11/18 by Teng Zhang, Zhang, Teng · 1 citation
Computer Science · Mathematics · #FOS: Mathematics #Mathematical Inequalities and Applications #Matrix Theory and Algorithms #Point processes and geometric inequalities #Spectral Theory (math.SP) #math.SP

paper · pdf · doi:10.48550/arxiv.1411.5058

openalex publication_date 2014/11/18 · arxiv created 2018/11/21 · arxiv updated 2018/11/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note proves the following inequality: if n=3k for some positive integer k, then for any n positive definite matrices A₁,A₂,⋯,A_n, (1)/(n³)‖∑_j1,j2,j3=1ⁿA_j1A_j2A_j3‖ ≥ ((n-3)!)/(n!) ‖∑_\substackj₁,j₂,j₃=1,
j₁, j₂, j₃ all distinctⁿA_j1A_j2A_j3‖, where ‖⋅‖ represents the operator norm. This inequality is a special case of a recent conjecture by Recht and Ré.

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