On biunimodular vectors for unitary matrices

TL;DR

This paper extends the study of biunimodular vectors from Fourier matrices to all unitary matrices, providing insights into their structure and broadening harmonic analysis applications.

Contribution

It introduces a general framework for finding biunimodular vectors in any unitary matrix, expanding beyond previous focus on Fourier matrices.

Findings

Biunimodular vectors exist for a wide class of unitary matrices.

The structure of unitary matrices can be better understood through biunimodular vectors.

The approach offers new tools for harmonic analysis and related fields.

Abstract

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.