= Cayley transform of a Hermitian matrix
{c}
{title2=$U=(A-iI)(A+iI)^{-1}$}
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 $U^\dagger U=I$. 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.
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