Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 3 / 40C / a / Solution

The power method starts with a unit vector ${\mathbf{x}}_0$ and iteratesIts eigenvalue estimate is the Rayleigh quotient

By the real spectral theorem, choose an orthonormal eigenbasis ${\mathbf{v}}_1,\dots ,{\mathbf{v}}_n$ and writeAssume $|{\lambda }_1|>|{\lambda }_2|$, ${\lambda }_1\ne 0$, and $c_1\ne 0$. Before normalization,Thus the direction error is $O(|{\lambda }_2/{\lambda }_1{|}^k)$. Orthogonality makes the linear terms vanish from the Rayleigh quotient:ConsequentlyThis is the quadratic Rayleigh-quotient improvement for the power method.