 Cayley transform (Lie theory)

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 $X^{†}J+JX=0$, expanding both sides givesso $C(X{)}^{†}JC(X)=J$. 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, $X=\mathrm{diag}(1/4,1/4,-1/2)$ 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.