Solution
= Solution
Expand the initial <vector> in the orthonormal <eigenbasis>. After $j$ unshifted steps, the coefficient ratio of mode $i<n$ to the dominant mode is $(c_i/c_n)(\lambda_i/\lambda_n)^j$. Since $\lambda_{n-1}=1$ and $c_{n-1},c_n\ne0$, the sharp asymptotic contraction factor is
$$
\boxed{\rho=\frac1{1+\epsilon}=1-\epsilon+O(\epsilon^2).}
$$
Normalization only changes the dominant coefficient by a higher-order error; its eventual sign is the sign of $c_n$.