The derivative of the matrix exponential

Let A be an n \times n complex matrix and define the mapping t \mapsto \mathsf{exp}(At) as usual. Prove that \frac{d}{dt} \mathsf{exp}(At) = A \mathsf{exp}(At).


Using this previous exercise, we have \frac{d}{dt} \mathsf{exp}(At) = \frac{d}{dt} \sum_k \frac{1}{k!} (At)^k = A \sum_k \frac{1}{(k+1)!} (k+1) (At)^k = A \sum_k \frac{1}{k!} (At)^k = A \mathsf{exp}(At) as desired.

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