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

For a skew-symmetric matrix, $M^T=-M$. Invariance of the determinant under transpose givesFor odd $n$, $2\det M=0$, so $\det M=0$ over the real or complex numbers.

Since $M$ is real, $(M^2{)}^T=(M^T{)}^2=(-M{)}^2=M^2$. For every real vector $x$,Thus $M^2$ is a real symmetric matrix and is negative semidefinite. When $n$ is odd, part (i) supplies a nonzero $x$ in the kernel of $M$, and then $M^{2}x=0$. Hence zero is an eigenvalue of $M^2$. In fact $\ker M^2=\ker M$, as the displayed norm identity shows.

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.

Normalize $e_1$ and put $f_1=M{e}_1/{\mu }_1$. The two vectors are an orthonormal basis of a plane $E$ invariant under $M$, since $M{e}_1={\mu }_1{f}_1$ and $M{f}_1=-{\mu }_1{e}_1$. Its orthogonal complement is also invariant: for $w\perp E$ and $z\in E$,Apply the spectral theorem for real symmetric matrices to the restriction of $M^2$ to $E^{\perp }$. Any remaining negative eigenvalue supplies another orthonormal pair by the same construction. Induction leaves only the kernel. Thus every nonzero eigenvalue of $M^2$ has even multiplicity. The pairs can share an eigenvalue, so this does not assert distinctness of the different $-{\mu }_j^2$.

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.