Cayley transformation of a skew-symmetric matrix is orthogonal?

Question

If $S$ is skew-symmetric ($S^{T} = -S$), how do I show that $Q$ is orthogonal where

$$Q = (I + S)(I - S)^{-1}$$ which is the Cayley transformation of $S$.

Answer 1

Just compute

$$QQ^T=(I+S)(I-S)^{-1}(I+S)^{-1}(I-S)$$ and since $(I+S)=-(I-S)+2I$ commutes with $(I-S)^{-1}$ then the result follows easily.

Remarks

1. We used the result $(A^{-1})^T=(A^T)^{-1}$
2. Prove for a skew-symmetric matrix S that $\pm1$ are not eigenvalues for it so that the inverse of $(S-I)$ and $(S+I)$ exist.