On biunimodular vectors for unitary matrices

A biunimodular vector of a unitary matrix \(A \in U(n)\) is a vector \(v \in \mathbb{T}^n\subset\bc^n\) such that \(Av \in \mathbb{T}^n\) as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object of extensive research in various areas of mathematics...

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Bibliographic Details
Published in:arXiv.org 2015-06
Main Authors: Führ, Hartmut, Rzeszotnik, Ziemowit
Format: Article
Language:English
Subjects:
Fourier analysis
Harmonic analysis
Mathematical analysis
Matrix algebra
Matrix methods
Vectors (mathematics)
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Summary:A biunimodular vector of a unitary matrix \(A \in U(n)\) is a vector \(v \in \mathbb{T}^n\subset\bc^n\) such that \(Av \in \mathbb{T}^n\) as well. Over the last 30 years, the sets of biunimodular vectors for Fourier matrices have been the object of extensive research in various areas of mathematics and applied sciences. Here, we broaden this basic harmonic analysis perspective and extend the search for biunimodular vectors to arbitrary unitary matrices. This search can be motivated in various ways. The main motivation is provided by the fact, that the existence of biunimodular vectors for an arbitrary unitary matrix allows for a natural understanding of the structure of all unitary matrices.
ISSN:2331-8422