Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 41B (Numerical Analysis)

The power method repeatedly normalizes $A{q}^{(k)}$ and estimates the eigenvalue with the Rayleigh quotient $(q^{(k)}{)}^{T}A{q}^{(k)}$. It converges when the dominant eigenvalue is unique in modulus and the initial vector has a nonzero component in its eigendirection.

The real spectral theorem gives real eigenvalues and orthogonal distinct eigenspaces. Constructively, a Householder reflection sends an eigenvector to $e_1$; symmetry makes the remaining block symmetric. Induction diagonalizes that block, producing orthogonal $Q$ with $Q^{T}AQ=\Lambda$. Multiplying by $Q$ gives $AQ{e}_k={\lambda }_{k}Q{e}_k$.

Write $q^{(0)}={\sum }_i{c}_i{q}_i$ with $c_1\ne 0$. After $k$ multiplications, the tangent of the angle to $q_1$ is at mostSince $|\sin {\theta }_k|\le \tan {\theta }_k$, the first bound follows. AlsoBound the coefficient by ${\max }_{i\ge 2}|{\lambda }_i-{\lambda }_1|$ and the sum by ${\sin }^2{\theta }_k$, then use the squared first estimate to obtain the second bound.