Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 39A / i / 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 $c_{n-1},c_n\ne 0$, the sharp asymptotic contraction factor isNormalization only changes the dominant coefficient by a higher-order error; its eventual sign is the sign of $c_n$.