Why is the notation for the inner product of two columns of the Vandermonde matrix expressed this way?

Question

I don't see how the above is a dot product for a Vandermonde matrix.

Answer 1

It is a dot product for two columns of a Vandermonde matrix, not the entire matrix. If one column is $(1, \omega^j, \omega^{2j}, \omega^{3j}...$, and say x = $\omega^j$, then that column is the more familiar (1, x, $x^2, x^3, ...)$ and another column likewise with $\omega^k$, then taking the complex transpose of the second vector gives a row vector (1, $\omega^{-k}, \omega^{-2k}$, ...) and then you take the dot product to get the result.