Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2013 / Paper 1 / 6 / iii / Solution

Choose $e_1$ real, as permitted by the spectral theorem for real symmetric matrices. Matrix powers commute, soThe proposed partner $e_2=M{e}_1$ is nonzero becauseAlso the real scalar $e_1^{T}M{e}_1$ equals its own transpose $e_1^T{M}^T{e}_1=-e_1^{T}M{e}_1$, so it vanishes. Therefore $e_1$ and $e_2$ are orthogonal eigenvectors with the same eigenvalue. This is the basic pairing in the paired eigenspaces of the square of a real skew-symmetric matrix.