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On Matrix Rearrangement Inequalities

by R. Alaifari and X. Cheng and L.B. Pierce and S. Steinerberger

(Report number 2019-65)

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 [10] 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.

Keywords: Rearrangement Inequality, Linear Operators, Matrix inequalitie

@Techreport{ACPS19_869,
  author = {R. Alaifari and X. Cheng and L.B. Pierce and S. Steinerberger},
  title = {On Matrix Rearrangement Inequalities},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-65},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-65.pdf },
  year = {2019}
}

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