= Inverse Cayley transform of a unitary matrix without eigenvalue one
{title2=$B=i(I+U)(I-U)^{-1}$}
If $U$ is a <unitary matrix> with no <eigenvalue> equal to one, the displayed inverse <Cayley transform of a Hermitian matrix> is Hermitian. Since $U^\dagger=U^{-1}$, taking the <Hermitian conjugate> gives $B^\dagger=-i(I-U^{-1})^{-1}(I+U^{-1})=i(I-U)^{-1}(I+U)=B$, using commutativity of rational functions of $U$. An eigenvalue $e^{i\theta}\ne1$ 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.
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