Given an invertible Vandermonde matrix $\mathbb{V}$, with randomly chosen knots $(v_0,v_1 \dots v_{N-1})$ on the unit circle, what can be said about the matrix:
$$\mathbb{A}= \mathbb{V}^{-1}. diag(v_0^N,v_1^N \dots, v_{N-1}^N) . \mathbb{V} $$
I know that it can be expressed as the $n^{th}$ power of the companion matrix corresponding to the roots. Does it have any special properties as such, or asymptotically?
