I'm stuck on the $3$rd part of a question, the first part was proving $A-iI$ is invertible when $A$ is Hermitian, second was proving $U=(A-iI)(A+iI)^{−1}$ is unitary. Now I'm being asked to prove $U-I$ is invertible and I'm having a hard time. All help would be appreciated.
2026-03-26 01:11:57.1774487517
If $A$ is Hermitian and $U=(A-iI)(A+iI)^{−1}$ is unitary, then $U-I$ is invertible
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$$ \begin{align} U-I&=(A-iI)(A+iI)^{-1}-I \\ &=(A-iI)(A+iI)^{-1}-(A+iI)(A+iI)^{-1} \\ &=(A-iI-A-iI)(A+iI)^{-1} \\ &= -2iI(A-iI)^{-1} \end{align} $$