= Cayley transform of a real skew-symmetric matrix
{c}
{title2=$Q=(I+A)(I-A)^{-1}$}
For a real <skew-symmetric matrix> $A$, $I-A$ is invertible: $(I-A)x=0$ would imply $\|x\|^2=x^TAx=0$. Its <Cayley transform (Lie theory)> is an <orthogonal matrix>, since transposition gives $Q^T=(I+A)^{-1}(I-A)$ and the commuting factors cancel in $Q^TQ$. Also $\det(I+A)=\det((I+A)^T)=\det(I-A)$, so $\det Q=1$ in every dimension. If $Av=\lambda v$, then $Qv=(1+\lambda)(1-\lambda)^{-1}v$. For $\lambda=it$, the transformed <eigenvalue> is $e^{2i\arctan t}$. In odd dimension $\det A=\det(-A)=-\det A$, so $A$ has a nonzero <kernel> and $Q$ fixes that direction.
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