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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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