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.

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