What is the value of (Vandermonde matrix)$^4~$?

Question

This is Vandermonde matrix:

$\omega$ is primitive N-root.

What is $W^4$?
I'm trying to calculate it but I don't know how exactly....

Thank you!

Answer 1

Firstly when u say $\omega$ is a primitive N-root, it means it is a root of $X^{N-1}+X^{N-2}+...X+1$ and so is $\omega^{2},...,\omega^{N-1}$.
First try to see what $W^2$ is, what will be the $i+1,j+1^{th}$ element of $W^2$
$\frac{1}{N}(1+\omega^{i+j}+\omega^{2(i+j)}+...+\omega^{(N-1)(i+j)})=0,$ if $i+j\neq0, N, 2N$ and $=1$ if $i=j=0$ or $i+j=N$ or $2N$. So,
$$W^2=\begin{bmatrix} 1&0& \dots &0 & 0 & 1 \\ 0 &0 & \dots&0&1&0\\ \vdots& \vdots& ⋰&\vdots\\ 1&0&\dots&0&0&1 \end{bmatrix}$$ Therefore,

$$W^4=\begin{bmatrix} 2 & 0 & \dots&0 &2 \\ 0 & 1 & \dots&0 &0\\ \vdots& \vdots& \ddots&\vdots&\vdots\\ 0&0&\dots&1&0\\ 2 & 0 & \dots&0 &2 \end{bmatrix}$$