# fourier matrix

Defined as

\begin{equation}
F = \frac{1}{\sqrt{N}} \cdot e^{\frac{-2\pi i mn}{N}}
\end{equation}

For all \(m, n\) from 0 to \(N-1\).

this is also the Vandermonde matrix for the roots of unity.

the inverse Fourier transform differs only in that the sign of the exponents are different.

An important property of the DFT matrix is that the first row (before scaling) will be all ones. This means that the first component of any signal, after applying the Fourier transform, will actually be the average of the signal itself. This is usually referred to as the DC component, and represents information about the signal from 0 in the signal.

The DFT matrix has some symmetry, that can be exploited in the FFT.