Past exam of the mathematical methods I course of the University of Oxford / 2013 / 8 / Solution

The second coordinate is already an eigenvector with eigenvalue $-2$. On the first and third coordinates the characteristic polynomial isThus the full eigenvalues are $3,-2,-2$. A corresponding orthonormal eigenbasis isTheir pairwise dot products vanish and each has norm one. Substitution checks $A{\mathbf{v}}_3=3{\mathbf{v}}_3$ and $A{\mathbf{v}}_{-2}=-2{\mathbf{v}}_{-2}$. The two-dimensional repeated eigenspace permits any orthonormal pair spanning the same plane; the displayed choice is convenient because it separates the second coordinate.