Derivative of Matrix Exponential
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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;
$\blacksquare$