Derivative of Matrix Exponential

Theorem
Let $\mathbf A$ be a square matrix.

Let $t \in \R$ be a real number.

Let $e^{\mathbf A t}$ denote the matrix exponential of $\mathbf A$.

Then:
 * $\dfrac \d {\d t} e^{\mathbf A t} = \mathbf A e^{\mathbf A t}$

Proof
From the definition of the matrix exponential, $e^{\mathbf A t}$ is defined as being the square matrix $X$ with the properties:


 * $(1): \quad \map {\dfrac \d {\d t} } X = \mathbf A X$
 * $(2): \quad \map X {\mathbf 0_n} = \mathbf I_n$

The result follows directly;