= Cayley transform
{disambiguate=Lie theory}
{c}
{title2=$C(X)=(I+X)(I-X)^{-1}$}
{wiki}
The Cayley transform is the rational <matrix> function $C(X)=(I+X)(I-X)^{-1}$, defined when $I-X$ is invertible. For the <orthogonal group>, <unitary group>, and analogous groups preserving a nondegenerate <bilinear form> or <Hermitian form>, it gives a local parametrization near the identity by their <Lie algebras>. If $X^\dagger J+JX=0$, expanding both sides gives
$$
(I+X)^\dagger J(I+X)=(I-X)^\dagger J(I-X),
$$
so $C(X)^\dagger JC(X)=J$. The inverse chart is $X=(Q-I)(Q+I)^{-1}$ wherever $Q+I$ is invertible. The <Cayley transform of a real skew-symmetric matrix> is the positive-definite real orthogonal example.
The formula is not a parametrization of an arbitrary <Matrix Lie group>. For example, $X=\operatorname{diag}(1/4,1/4,-1/2)$ belongs to the <special linear Lie algebra>, but $\det C(X)=25/27$, so its transform is not in the <special linear group>. In indefinite form-preserving groups, the Cayley chart's full domain can also meet components outside the <identity component of a Lie group>; its local behavior near zero alone does not imply a global parametrization.
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