= Cross-product matrix spectrum
{title2=$\operatorname{spec}([n]_\times)=\{0,i|n|,-i|n|\}$}
The <cross-product matrix> $A\mathbf x=\mathbf n\times\mathbf x$ obeys $A\mathbf n=0$ and $A^2=\mathbf n\mathbf n^T-|\mathbf n|^2I$, by the <vector triple product>. On the <orthogonal complement> of a nonzero $\mathbf n$, this is $-|\mathbf n|^2I$; hence the two remaining complex <eigenvalues> are $\pm i|\mathbf n|$. In an orthonormal basis adapted to $\mathbf n$, $A$ is a planar quarter-turn multiplied by $|\mathbf n|$, with zero action along its axis.
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