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 $(1-i\lambda )/(1+i\lambda )$.