## A class of group homomorphisms from CC to GL(n,CC)

Fix an $n \times n$ matrix $M$ over $\mathbb{C}$. Define a mapping $\psi_M : \mathbb{C} \rightarrow \mathsf{GL}_n(\mathbb{C})$ by $\alpha \mapsto \mathsf{exp}(M\alpha)$. Prove that $\psi_M$ is a group homomorphism on $(\mathbb{C},+)$.

Note that $\mathsf{exp}(M\alpha)$ is nonsingular by this previous exercise, so that $\psi_M$ is properly defined.

Now $\psi_M(\alpha + \beta) = \mathsf{exp}(M(\alpha+\beta))$ $= \mathsf{exp}(M\alpha + M\beta)$ $= \mathsf{exp}(M\alpha) \cdot \mathsf{exp}(M\beta)$ by this previous exercise, which equals $\psi_M(\alpha) \psi_M(\beta)$. So $\psi_M$ is a group homomorphism.