= Paired eigenspaces of the square of a real skew-symmetric matrix
{title2=$\max\#\operatorname{spec}(M^2)=\lceil n/2\rceil$}
For a real <skew-symmetric matrix> $M$, $M^2$ is symmetric and $x^TM^2x=-\|Mx\|^2$. On an <eigenspace> with <eigenvalue> $-\mu^2<0$, every unit <vector> $u$ has unit partner $v=Mu/\mu$, orthogonal to $u$, with $Mu=\mu v$ and $Mv=-\mu u$. Its plane and its <orthogonal complement> are invariant under $M$. Repeating gives real orthogonal two-dimensional blocks $\begin{pmatrix}0&-\mu\\\mu&0\end{pmatrix}$ and a zero block. Thus every nonzero <eigenvalue> of $M^2$ has even <multiplicity>, the zero multiplicity has the parity of $n$, and the number of distinct <eigenvalues> is at most $n/2$ for even $n$ and $(n+1)/2$ for odd $n$. Distinct positive block parameters attain the bound, with one zero direction added when $n$ is odd.
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