Consider a sequence $x = \{ x_n \} _{n=0} ^{N-1}$ with discrete fourier transform denoted by $X = \{X_k \}_{k=0}^{N-1}$. If $x = (1, -i , 2, 3)$ then find the matrix which maps $x$ to $X$ and compute $X$ to find the discrete fourier transforms of $\overline{x}, \text{Re} (x),$ and $\text{Im} (x)$. (Here, the overline denotes pointwise complex conjugate.)
The only thing I can think of is to use the Vandermonde matrix for the roots of unity, up to the normalization factor, but I do not see how this would give such a matrix. Is there another way to go about this or any ideas on how to proceed? There is a somewhat similar question here that analyzes $\overline{x}$ only, but it does not utilize matrices).