Exercise 4.1.5

With $N=3$, we have $w_3=e^{2\mathit{\pi i}∕3}=-\frac{1}{2}+\frac{\sqrt{3}}{2}i$

$F_3=\left[\begin{array}{ccc}\hfill w_3^{0\times 0}\hfill & \hfill w_3^{1\times 0}\hfill & \hfill w_3^{2\times 0}\hfill \\ \hfill w_3^{0\times 1}\hfill & \hfill w_3^{1\times 1}\hfill & \hfill w_3^{2\times 1}\hfill \\ \hfill w_3^{0\times 2}\hfill & \hfill w_3^{1\times 2}\hfill & \hfill w_3^{2\times 2}\hfill \\ \hfill \hfill \end{array}\right]=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill 1\hfill & \hfill w_3\hfill & \hfill w_3^2\hfill \\ \hfill 1\hfill & \hfill w_3^2\hfill & \hfill w_3^4\hfill \end{array}\right]$

The permutation matrix for the even-odd rows is

And the Matrix $D_3=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill w_6\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill w_6^2\hfill \end{array}\right]$

We have

Since we have $w_3=w_6^2,w_6^3=-1,w_3^4=w_3$, it’s easy to check that the $F_6$ is the correct matrix.