# Ways to calculate the derivative of the matrix exponential

Could someone provide me with a rigorous proof as to why the derivative of the function $f:t \ni \mathbb{R} \mapsto e^{tA}\in \textrm{Mat}_n (\mathbb{R})$ is $t \mapsto A\cdot e^{tA}$ ? I didn't understand the "elementwise" arguments, as to why the above should hold and when trying to evaluate $f'$ by hand I got stuck at evaluating $$\lim _{h\rightarrow 0} \frac{||e^{xA}\cdot e^{hA} - e^{xA} -A\cdot e^{xA} \cdot h||}{|h|},$$

where $||\cdot ||$ denotes any norm on $\textrm{Mat}_n (\mathbb{R})$ that is multiplicative (so that $( \textrm{Mat}_n (\mathbb{R}) , || \cdot ||)$ becomes a Banach algebra) - which is what I have do to, I think (please correct me, if I'm wrong, or using an unnecessary abstract level of discourse), because in the setting of matrix-valued functions the derivative of a matrix becomes the derivative between the Banach spaces $\mathbb{R}$ and $\textrm{Mat}_n (\mathbb{R})$ (using an isomorphism $\phi:\textrm{Mat}_n (\mathbb{R}) \rightarrow \mathbb{R}^{n^2}$ to do the derivative there and then transporting everything back to $\textrm{Mat}_n (\mathbb{R})$ seems rather ugly to me - although I tried to do it this way and failed).

Could I replace $\mathbb{R}$ with $\mathbb{C}$ ?

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There is a factor $h$ missing in the third term of the numerator of the limit formula. –  Raskolnikov Nov 11 '11 at 19:35
You can just replace $e^{tA}$ by series which converge absolutely because $\|e^{tA}\|\leq e^{t\|A\|}$, that may make things easier. –  Ilya Nov 11 '11 at 19:35
$e^{tA}$ is, by definition, just a matrix of power series in $\mathbb{R}$. To find the derivative, differentiate term-by-term. Which part of the proof do you consider in doubt? That the power series converges? That it is differentiable? –  user7530 Nov 11 '11 at 19:54
You can make life simpler by factoring out $e^{xA}$, so $$\frac{e^{(x+h)A}-e^{xA}}{h}=\frac{e^{hA}-I_n}h\;e^{xA}.$$ Then from the power series of $e^{hA}$ only the term linear in $h$ matters. –  Jyrki Lahtonen Nov 11 '11 at 20:43