# On matrix rearrangement inequalities

Journal Article (Journal Article)

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 AmBn? For example, is ∥ AABAABABB ∥ ≤ ∥AAAAABBBB∥? Drury [Electron J. Linear Algebra 18 (2009), pp. 13 20] 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.

### Full Text

### Duke Authors

### Cited Authors

- Alaifari, R; Cheng, X; Pierce, LB; Steinerberger, S

### Published Date

- January 1, 2020

### Published In

### Volume / Issue

- 148 / 5

### Start / End Page

- 1835 - 1848

### Electronic International Standard Serial Number (EISSN)

- 1088-6826

### International Standard Serial Number (ISSN)

- 0002-9939

### Digital Object Identifier (DOI)

- 10.1090/proc/14831

### Citation Source

- Scopus