= Skew-symmetric exponential as an axial rotation
{title2=$e^A\in\operatorname{SO}(3)\quad(A^T=-A)$}
For a nonzero real three-dimensional <skew-symmetric matrix>, choose real orthogonal vectors $u,v$ of equal length with $Au=\theta v$, $Av=-\theta u$ and $\theta>0$, and a real unit vector $e_3$ in its kernel. The <matrix exponential> acts by $e^Au=u\cos\theta+v\sin\theta$, $e^Av=v\cos\theta-u\sin\theta$ and $e^Ae_3=e_3$. Its matrix in this <orthonormal basis> is a planar <rotation matrix> with an unchanged axis. A purely imaginary nonzero <eigenvalue> and its complex <eigenvector> give $u,v$ by taking real and imaginary parts; antisymmetry proves their equal norms and orthogonality.
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