Exercise 2.5

Let $S\in {ℂ}^{m\times n}$ be skew-hermitian, i.e., $S^{\text{∗}}=-S$.
(a) Show by using Exercise 2.1 that the eigenvalues of $S$ are pure imaginary. (b) Show that $I-S$ is nonsingular.
(c) Show that the matrix $Q={(I-s)}^{-1}(I+S)$, known as the Cayley transform of $S$, is unitary. This is a matrix analogue of a linear fractional transformation $(1+s)∕(1-s)$, which maps the left half of the complex $s$-plane conformally onto the unit disk.)