The Fourier matrix is given by
where $\omega = e^{-2\pi i/N}$. Is there any clever way to calculate Frobenius norm of Fourier matrix?
I tried solving it with brute force and got some ugly calculations
2026-03-26 22:14:01.1774563241
Frobenius norm of Fourier matrix
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This is a straightforward computation using any reasonable definition of the Frobenius norm. For instance, we have $$ \|W\| = \sqrt{\operatorname{tr}(W^*W)} = \sqrt{\operatorname{tr}(I)} = \sqrt{N} $$ where $W^*W = I$ since $W$ is a unitary matrix.