Exercise 14, § 9

(a) Let $G = \mathbf{GL}(n, \mathbf{C})$. Show that $\exp_G$ is surjective. (Use the holomorphic functional calculus of Spectral Theories, Chapter I, § 4, no. 8.)

(b) Let $G' = \mathbf{SL}(2, \mathbf{C})$. Show that, if $\sigma \in \mathbf{C}^*$, the element $\begin{pmatrix} -1 & \sigma \\ 0 & -1 \end{pmatrix}$ of $G'$ does not belong to the image of $\exp_{G'}$.