  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.
