Proof of the Cayley Transform. Showing that $H=i(I-U)(I+U)^{-1}$ where $U=(I+iH)(I-iH)^{-1}$ for a Hermitian matrix $H$.

Question

I am working on showing that a Hermitian matrix $H=i(I-U)(I+U)^{-1}$ where $U=(I+iH)(I-iH)^{-1}$.

I have already shown that $I-iH$ is invertible and that $U$ is unitary, but am stuck on showing that $U$ can be written in this form. Any help would be greatly appreciated !

Thank you so much!

Answer 1

If you start with $U=(I+iH)(I-iH)^{-1}$, and you want to find $H$ in terms of $U$, start by multiplying by sides on the right by $I-iH$: $$ U(I-iH)=(I+iH) \\ U-iUH = I+iH \\ U-I = i(U+I)H \\ -i(U-I)(U+I)^{-1}=H\\ H=i(I-U)(I+U)^{-1}. $$