Solution
= 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 $Q^TAQ=\Lambda$. Multiplying by $Q$ gives $AQe_k=\lambda_kQe_k$.