Given $\exp(tA)$, find $A$

Question

Given $$\exp(tA)=e^t\begin{pmatrix}\cos t-\sin t&-\sin t\\2\sin t&\cos t+\sin t \end{pmatrix},$$ is there a way to construct $A$? The only Information I can think of right now that is easily obtainable is the trace $\text{tr} A$ through $\det\exp(tA)=e^{\text{tr}A}$. In this case $\text{tr}A=2$. But how do I get i.e. the eigenvalues?

Answer 1

It is not hard:$$A=\left.\frac{\mathrm d}{\mathrm dt}\right|_{t=0}\exp(tA).$$

Answer 2

$$ e^{tA} = I + tA + \frac {t^2}{2!}A^2 + \frac {t^3}{3!}A^3 +....$$

$$ \frac {d}{dt} e^{tA} = A + {t}A^2 + \frac {t^2}{2!}A^3 +....$$

Let $t=0$ and you get $$ \frac {d}{dt} e^{tA}\big |_{t=0} = A$$

That is differentiate $e^{tA}$ and evaluate the result at $t=0$ to get $A$