This is the Spectral Theorem. It says (in the version for real matrices, which are what we have here) that any symmetric matrix has an orthonormal basis of eige

This is the Spectral Theorem. It says (in the version for real matrices, which are what we have here) that any symmetric matrix has an orthonormal basis of eigenvectors, or equivalently that any symmetric matrix can be diagonalized via an orthogonal change of coordinates.