Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2016 / Paper 1 / 14 / b / i

The third coordinate is decoupled, giving the unit eigenvector $(0,0,1{)}^{\mathsf{T}}$ with eigenvalue $1$. In the first two coordinates, the symmetric direction $(1,1)$ has eigenvalue $3-1=2$, and the antisymmetric direction $(1,-1)$ has eigenvalue $3+1=4$. Their normalized forms areThey are an orthonormal basis of ${\mathbb{R}}^3$; independent sign reversals are equally valid choices.