[math] \newcommand{\mathds}{\mathbb}[/math]

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Prove that the flat matrix, which is the all-one [math]N\times N[/math] matrix, diagonalizes over the complex numbers as follows,

[[math]] \begin{pmatrix} 1&\ldots&\ldots&1\\ \vdots&&&\vdots\\ \vdots&&&\vdots\\ 1&\ldots&\ldots&1\end{pmatrix}=\frac{1}{N}\,F_N \begin{pmatrix} N\\ &0\\ &&\ddots\\ &&&0\end{pmatrix}F_N^* [[/math]]

where [math]F_N=(w^{ij})_{ij}[/math] with [math]w=e^{2\pi i/N}[/math] is the Fourier matrix, with the convention that the indices are taken to be [math]i,j=0,1,\ldots,N-1[/math].