= Solution
Put $D=|\langle L_N,f\rangle-\langle\widehat L_N,f\rangle|$. Apply the <Cauchy-Schwarz inequality> to the average of the nonnegative <eigenvalue> differences, and then the <Hoffman–Wielandt inequality>:
$$
D^2\leq\left(\frac1N\sum_i|\lambda_i-\widehat\lambda_i|\right)^2\leq\frac1N\sum_i|\lambda_i-\widehat\lambda_i|^2\leq\frac1N\operatorname{Tr}[(X_N-\widehat X_N)^2].
$$
Taking nonnegative square roots gives the <spectral Lipschitz bound from Frobenius distance>
$$
\boxed{D\leq\left(\frac1N\operatorname{Tr}[(X_N-\widehat X_N)^2]\right)^{1/2}=\frac{\|X_N-\widehat X_N\|_F}{\sqrt N}.}
$$
The factor $N^{-1/2}$ is essential: the <empirical spectral measure> has total mass $1$, not $N$.
Back to article page
