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

Collect the paired orthonormal bases from part (iv), followed by an orthonormal basis of $\ker M^2=\ker M$. The matrix $Q$ of these columns is an orthogonal matrix, andIndeed, each pair obeys $M{e}_j={\mu }_j{f}_j$, $M{f}_j=-{\mu }_j{e}_j$, and all kernel vectors are annihilated. This proves the real block form of a skew-symmetric matrix directly, including repeated block parameters. In particular its rank is $2r$, always even.