Exercise 4.2.6

The eigenvalues of $C$ is $\mathit{Fc}$, when $\mathit{Fc}$ has no zeroes, that means no zero eigenvalues, so the matrix $C$ is invertible.

${\sum <!--\; FUNCTION\; APPLICATION\; -->}_0^{N-1}{c}_j{e}^{\mathit{ij𝜃}}=\left[\begin{array}{cccc}\hfill 1\hfill & \hfill e^{\mathit{𝜃i}}\hfill & \hfill \dots \hfill & \hfill e^{(N-1)\mathit{𝜃i}}\hfill \end{array}\right]\left[\begin{array}{c}\hfill c_0\hfill \\ \hfill c_1\hfill \\ \hfill ⋮\hfill \\ \hfill c_{N-1}\hfill \end{array}\right]$

When $𝜃=\frac{2\pi }{N}$, we have $e^{\mathit{𝜃i}}=e^{2\mathit{\pi i}∕N}=w$, so the sum is exactly the dot product of each row of $F$ with $c$. And we want each dot product to be nonzero, exactly as required in the statement.