$A$ is skew hermitian, prove $(I-A)^{-1} (I+A)$ is unitary Given $A$ is a skew-hermitian, (i.e $A^H=−A$), the Cayley transform of $A$ is defined as: $W=(I-A)^{-1} (I+A)$. How can be proved that $W$ is unitary (i.e. $W^H W = W W^H = I$)?

I present a slight modification on the answer to a very similar question. I assume we are working over the complex numbers. 
Recall the eigenvalues of a skew-Hermitian matrix are imaginary. So the set of eigenvalues of $J=I-A$ cannot contain zero, and the matrix is invertible. We then have $W=J^{-1} J^H$. it remains to compute:
$$J^{-1} J^H (J^{-1} J^H)^H= J^{-1} J^H J (J^{-1})^H = J^{-1} J^H J (J^{H})^{-1}$$
Note now that $J$ and $J^H$ commute.
$$J^{-1} J^H J (J^{H})^{-1}=J^{-1} J J^H (J^{H})^{-1}=I.$$
The other computation is almost exactly the same.
