\begin{equation}
\label{eq:rate}
\tan\angle(U_k,AX)
\leq\frac{\lvert\lambda_{\ell+1}\rvert}{\lvert\lambda_k\rvert}
\tan\angle(U_k,X)
+\frac{\bigl\lVert\Lambda_{\ell\backslash k}X_{\ell\backslash k}
X_k^{-1}\bigr\rVert_2}{\lvert\lambda_k\rvert}.
\end{equation}
