# On Matrix Rearrangement Inequalities

## TL;DR

This paper proves that matrix rearrangement inequalities hold for all disordered words in 2x2 matrices and for most small perturbations of the identity in larger matrices, extending previous partial results.

## Contribution

It establishes the validity of matrix rearrangement inequalities for all disordered words in 2x2 matrices and for generic small perturbations in larger matrices, improving upon prior characterizations.

## Key findings

- Rearrangement inequality holds for all disordered words in 2x2 matrices.
- For larger matrices, the inequality holds for most small perturbations of the identity.
- Counterexamples exist only for specific matrix sizes and configurations.

## 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 $A^m B^n$? For example, is $$ \| AABAABABB \| \leq \| AAAAABBBB \|\ ? $$ 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 $3 \times 3$ matrices, and thus as stated the characterization applies only for $N \times N$ matrices with $N \geq 3$. In contrast, we prove that for $2 \times 2$ matrices, the general rearrangement inequality holds for all disordered words. We also show that for larger $N \times N$ 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.