Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2014 / Paper 1 / 6 / i / Solution

The spectral theorem for real symmetric matrices supplies real eigenvalues and an orthonormal eigenbasis. Compute eigenvectors, normalize them, and within each repeated-eigenvalue eigenspace choose an orthonormal basis. Distinct eigenspaces are already orthogonal because symmetry gives $(\lambda -\mu )v^{T}w=0$ for eigenvectors with eigenvalues $\lambda ,\mu$.

Use the resulting basis vectors as columns of $O$. ThenThe order and signs of the columns can be chosen freely; rotations within a repeated-eigenvalue eigenspace are also allowed.