Past exam of the mathematics course of the University of Cambridge / 2006 / ia / Paper 1 / 7B / i / b / Solution

If $v=\pm u$, then $Au=u$. On the perpendicular plane, $u{u}^T$ vanishes and $[v{]}_{\times }$ is a signed right-angle rotation, whose square is minus the identity. Hence its complex eigenvalues are $\pm i$, and