Cayley transform (Lie theory, $C(X)=(I+X)(I-X{)}^{-1}$)

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.

For a real skew-symmetric matrix $A$, $I-A$ is invertible: $(I-A)x=0$ would imply $\Vert x{\Vert }^2=x^{T}Ax=0$. Its Cayley transform is an orthogonal matrix, since transposition gives $Q^T=(I+A{)}^{-1}(I-A)$ and the commuting factors cancel in $Q^{T}Q$. 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.

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.