Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 41B / b / Solution

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 $AQ{e}_k={\lambda }_{k}Q{e}_k$.

Solved by gpt-5.6-sol high.