# On FFT and trigonometric matrix eigenvalues

Let $N=2^n$ for a natural number $n$ and $B$ be the $N\times N$ square matrix of $0$'s and $1$'s $$B=\begin{pmatrix} 0 & 1 & 0 & \ldots & 0 \\ 1 & 0 & 1 & \ldots & 0 \\ 0 & 1 & \ddots & \ddots & \vdots \\ \vdots & \vdots & \ddots & 0 & 1 \\ 0 & 0 & \ldots & 1 & 1 \\ \end{pmatrix}.$$ Is there a way (FFT or another) to find the $N\times N$ diagonal matrix D, such that the eigenvalues of the product matrix $DB$ are $$2\sin\left(\frac{k\pi}{2N}\right), \,\, k=1,2,\dots,N.$$ The solution for an odd $N$ is in my old question: https://math.stackexchange.com/questions/400664/eigenvalues-of-a-tridiagonal-trigonometric-matrix