Shifted Fourier Matrices and Their Tridiagonal Commutors

It is known that, for $ n \ge 3 $, the n × n Fourier matrix $ {F} = n^{-1/2} [e^{2 \pi {{\text{\textit{\r{\i}}}}} \mu \nu/n}] $ ($ 0 \le \mu < n $, $ 0 \le \nu < n $) commutes with a nonscalar tridiagonal matrix T and also with another matrix X that is "almost" tridiagonal. These matrices are important in selecting eigenvectors for the Fourier matrix itself. The purpose of this paper is to generalize those results to matrices $ F_n({\tau},{{\alpha}}) $, variants of the Fourier matrix depending on a base-choosing parameter $\tau$ and a shift parameter $\alpha$. These \emph{shifted Fourier matrices} are defined by $ F_n({\tau},{{\alpha}}) = %\fffnta{n}{\tau}{\alfa} = n^{-1/2} [e^{2\pi{{\text{\textit{\r{\i}}}}} \tau (\mu - m + a)(\nu - m + a)}] $, where $ {m} = (n - 1)/2 $ and $ a = \alpha/\tau $. We show that $ F_n({\tau},{{\alpha}}) $ commutes with a nonscalar tridiagonal matrix $ T_n({\tau},{{\alpha}}) $ for all values of the shift parameter $\alpha$, as long as the base $ q = e^{2\pi{{\text{\textit{\r{\i}}}}} \tau} $ determined by $ \tau $ is an $n$th root of unity, and also for all values of $\tau$ in the "centered" case corresponding to $ \alpha = 0 $. Furthermore, we show that, in certain more specialized cases, $ F_n({\tau},{{\alpha}}) $ also commutes with a matrix $ X_n({\tau},{{\alpha}}) $ that has $ \pm 1 $ in the upper-right and lower-left corners and is otherwise tridiagonal. In most cases, $ T_n({\tau},{{\alpha}}) $ and $ X_n({\tau},{{\alpha}}) $ are essentially the only matrices of their band-structure that commute with $ F_n({\tau},{{\alpha}})$.