= Symmetric and skew-symmetric Cayley transforms
{title2=$C=(I-B)(I+B)^{-1}$}
For a real <skew-symmetric matrix> $B$, its <Cayley transform of a real skew-symmetric matrix> with the opposite sign convention has inverse $C^{-1}=(I+B)(I-B)^{-1}$. Since $C$ is an <orthogonal matrix>, symmetry means $C=C^{-1}$, or $(I-B)^2=(I+B)^2$. This forces $B=0$. Skew-symmetry means $C=-C^{-1}$, or $(I-B)^2=-(I+B)^2$, which is equivalent to $B^2=-I$. In that case $C=-B$. Such examples exist in even dimension by taking two-dimensional right-angle <rotation matrices> in diagonal blocks; in odd real dimension the determinant equation $(\det B)^2=(-1)^n$ rules them out.
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