## The Matrix Exponential of a Diagonal Matrix

## Problem 681

For a square matrix $M$, its **matrix exponential** is defined by

\[e^M = \sum_{i=0}^\infty \frac{M^k}{k!}.\]

Suppose that $M$ is a diagonal matrix

\[ M = \begin{bmatrix} m_{1 1} & 0 & 0 & \cdots & 0 \\ 0 & m_{2 2} & 0 & \cdots & 0 \\ 0 & 0 & m_{3 3} & \cdots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & m_{n n} \end{bmatrix}.\]

Find the matrix exponential $e^M$.

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