 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 $AQ{e}_k={\lambda }_{k}Q{e}_k$.