= Rank-one eigenprojector perturbation bound
{title2=$\|\widehat P-P\|_F\leq2\sqrt2\|W\|_{\mathrm{op}}/\gamma$}
Let $\widehat P$ project onto a leading unit <eigenvector> of $H+W$, with $H$ as in the <Davis-Kahan curvature lemma>. Its maximizing <Rayleigh quotient> gives $\operatorname{tr}(H(P-\widehat P))\leq\operatorname{tr}(W(\widehat P-P))$. Duality of the <operator norm> and <trace norm>, together with the rank-two bound $\|\widehat P-P\|_*\leq\sqrt2\|\widehat P-P\|_F$, proves the estimate. No gap condition on the perturbed matrix is needed.
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