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On matrix rearrangement inequalities


Authors: Rima Alaifari, Xiuyuan Cheng, Lillian B. Pierce and Stefan Steinerberger
Journal: Proc. Amer. Math. Soc. 148 (2020), 1835-1848
MSC (2010): Primary 15A45, 47A30, 47A63; Secondary 39B42
DOI: https://doi.org/10.1090/proc/14831
Published electronically: February 13, 2020
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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

$\displaystyle \Vert AABAABABB \Vert \leq \Vert AAAAABBBB \Vert ?$    

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 \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.

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Additional Information

Rima Alaifari
Affiliation: Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland
Email: rima.alaifari@math.ethz.ch

Xiuyuan Cheng
Affiliation: Department of Mathematics, Duke University, 120 Science Drive, Durham, North Carolina 27708
Email: xiuyuan.cheng@duke.edu

Lillian B. Pierce
Affiliation: Department of Mathematics, Duke University, 120 Science Drive, Durham, North Carolina 27708
Email: pierce@math.duke.edu

Stefan Steinerberger
Affiliation: Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, Connecticut 06511
Email: stefan.steinerberger@yale.edu

DOI: https://doi.org/10.1090/proc/14831
Keywords: Rearrangement inequality, linear operators, matrix inequalities.
Received by editor(s): April 17, 2019
Received by editor(s) in revised form: August 22, 2019, and August 25, 2019
Published electronically: February 13, 2020
Additional Notes: The second author was partially supported by the NSF (DMS-1818945, DMS-1820827).
The third author was partially supported by CAREER grant NSF DMS-1652173 and the Alfred P. Sloan Foundation.
The fourth author was partially supported by the NSF (DMS-1763179) and the Alfred P. Sloan Foundation.
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2020 American Mathematical Society