The matrix exponential is defined by the same formula as the scalar exponential, that is by the power series:

\$e^X = \frac{1}{0!} X^0 + \frac{1}{1!} X^1 + \frac{1}{2!} X^2 + \cdots = I + X + \frac{1}{2!} X^2 + \cdots\$

where \$$I\$$ is the identity matrix.