Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2011 / Paper 1 / 13 / a / iv / Solution

The chosen eigenvectors have dot productThey are therefore orthogonal, and their unit lengths make them an orthonormal eigenbasis. This agrees with the spectral theorem for real symmetric matrices, since $A$ is real symmetric and its two eigenvalues are distinct.