On Matrix Rearrangement Inequalities

2019/04/10 by Alaifari, Rima, Cheng, Xiuyuan, Pierce, Lillian B. +1
#FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1904.05239

Abstract

Given two symmetric and positive semidefinite square matrices A, B, is it true that any matrix given as the product of m copies of A and n copies of B in a particular sequence must be dominated in the spectral norm by the ordered matrix product A^m Bⁿ? For example, is ‖ AABAABABB ‖ ≤ ‖ AAAAABBBB ‖ ? Drury has characterized precisely which disordered words have the property that an inequality of this type holds for all matrices A,B. However, the 1-parameter family of counterexamples Drury constructs for these characterizations is comprised of 3 × 3 matrices, and thus as stated the characterization applies only for N × N matrices with N ≥ 3. In contrast, we prove that for 2 × 2 matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger N × N matrices, the general rearrangement inequality holds for all disordered words, for most A,B (in a sense of full measure) that are sufficiently small perturbations of the identity.

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