Cayley transform of a Hermitian matrix ($(I+iA{)}^{-1}(I-iA)$)

If $A$ is a Hermitian matrix, then $(I+iA{)}^{-1}(I-iA)$ is unitary. This Cayley transform maps each real eigenvalue $\lambda$ to the unit-modulus number $(1-i\lambda )/(1+i\lambda )$.

If $U$ is a unitary matrix with no eigenvalue equal to one, the displayed inverse Cayley transform of a Hermitian matrix is Hermitian. Since $U^{†}=U^{-1}$, taking the Hermitian conjugate gives $B^{†}=-i(I-U^{-1}{)}^{-1}(I+U^{-1})=i(I-U{)}^{-1}(I+U)=B$, using commutativity of rational functions of $U$. An eigenvalue $e^{i\theta }\ne 1$ is mapped to the real number $-\cot (\theta /2)$. Conversely $U=(B-iI)(B+iI{)}^{-1}$, so this correspondence identifies Hermitian matrices with unitary matrices whose spectrum avoids one.