Variations on a theme of Frobenius about almost commuting unitary matrices
type: Item
instance of: scholarly article
title: Variations on a theme of Frobenius about almost commuting unitary matrices
zbMATH Open document ID: 1072.15018
author: Marcel Steiner
published in: Linear Algebra and its Applications
publication date: 2005-06-01T00:00:00Z
review text: For two rotations \(A\) and \(B\) in \({\mathbb R}^n\), where \(B\) does not rotate any vector by an angle of \(\frac{\pi}{2}\) radius, \textit{F. G. Frobenius} [Sitzungsber. Akad. Wiss. Berlin, 654--665 (1911; JFM 42.0153.01)] proved that if \(A\) commutes with the commutator \( [ A,B ] = A^{-1}B^{-1}AB\) then \(A\) and \(B\) are commuting rotations. Here, the Frobenius' theorem is generalized for almost commuting unitary matrices. The proof of this generalized theorem is based on the fact that if two unitary matrices \(A\) and \(C\) almost commute, then there exists an explicit construction of a unitary change of the basis \(V\) so that \(V^{*}AV\) and \(V^{*}CV\) are simultaneously almost diagonal. The presented results with estimations of the error terms could be of interest in numerical matrix computations.
Mathematics Subject Classification ID: 15A27
zbMATH DE Number: 2173013
zbMATH Keywords: almost commuting unitary matrices
zbMATH Keywords: crystallographic group
MaRDI profile type: Publication
full work available at URL: j.laa.2004.12.001
OpenAlex ID: W1972752218
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DOI: 10.1016/J.LAA.2004.12.001
description: scientific article; zbMATH DE number 2173013