What is the formal inverse of a square $N\times N$ matrix $A$ with entries $A_{ij}=a^{(i-1)(j-1)}$?
When $a$ is the $N$th root of unity (i.e. $a=\exp(2 \pi i/N)$), then $A$ is the Fourier matrix and its inverse is simply the conjugate of $A$, but what happens if $a$ is not a root of unity?