Cayley transform of a Hermitian matrix ($U=(A-iI)(A+iI{)}^{-1}$)

For a Hermitian matrix $A$, both $A+iI$ and $A-iI$ are invertible because every eigenvalue of $A$ is real. The displayed rational function is a unitary matrix: taking its conjugate transpose and commuting the rational factors gives $U^{†}U=I$. It maps each eigenvector of $A$ with eigenvalue $\lambda$ to the same vector with eigenvalue $(\lambda -i)/(\lambda +i)$, a complex number of modulus one different from one. The inverse Cayley transform of a unitary matrix without eigenvalue one recovers $A$.

Another common convention is $C=(I+iA{)}^{-1}(I-iA)$, which sends $\lambda$ to $(1-i\lambda )/(1+i\lambda )$ and avoids eigenvalue minus one. The two conventions are related by $C=-U^{-1}$; the inverse formula must match the chosen convention.

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.