I'm interested in the $N \times N$ matrix $A$ with Dirichlet entries \begin{align} a_{n,m}=\frac{\sin \left(\pi\eta(n-m)\right)}{\sin \left(\pi\eta (n-m)/{N}\right)}. \end{align} I found that \begin{align} \frac{\text{squared sum of diagonal entries }(=N^3)}{\text{squared Frobenius norm of } A} \approx \eta \end{align} as $N$ becomes large, with numerical experiment. What approach should I try to prove this?
Thanks.
EDIT: I have derived it with early results on prolate matrices. It is well-known that, as $N$ becomes large, nearly $\eta N$ nonzero eigenvalues of $A$ are clustered to $N/\eta$. Hence \begin{align} \|A\|_F^2\approx (\eta N)\times (N/\eta)^2 =N^3/\eta. \end{align}