My professor assigned me this exercise:
Let $D\in \mathbb{C}^{n\times n}$ be a diagonal matrix
\begin{pmatrix} \lambda_1 & & \\ & \ddots & \\ & & \lambda_n \end{pmatrix}
for some $\lambda_1,...,\lambda_n \in \mathbb{C}$. Find $e^D$. Here in order to find $e^D$ I need to use this theorem:
Th. Let $X$ be a Banach space. If $A\in \mathcal{L}(X)$, then the series $e^A = \sum_{n=0}^\infty \frac{1}{n!}A^n$ converges in $\mathcal{L}(X)$ (here $\mathcal{L}(X)$ is Banach algebra).
Can somebody explain to me what do I need to do in this task? Maybe just calculate the determinant of D and find a limit of this series?