how to differential exponential of a matrix variable $f(X)=e^{X(t)\mathrm{d}t}$?

Question

I have a function about a square matrix $X$ which depends also time: $f(X)=e^{X(t)}$, $t$ is time. So how to differential it about time to have $\frac{\mathrm{d}f(X)}{\mathrm{d}t}$?

I know that $$e^{X(t)}=I+X(t)+\frac{X(t)^2}{2!}+\frac{X(t)^3}{3!}+\dots$$ But I dont know how to differential each term $\frac{X(t)^k}{k!}$ about time $t$. Or is there some other method to differential directly $e^{X(t)}$?

Any help will be appreciated. thanks.

Answer 1

There is no simple formula for the derivative of the exponential of a matrix valued function.
However, there is a well known integral representation of the derivative. $$\frac{d}{dt} e^{X(t)} = \int_0^1 e^{\lambda X(t)} \left(\frac{dX(t)}{dt}\right) e^{(1-\lambda)X(t)} d\lambda$$

Look at the wiki page for matrix exponential, there are some references for this integral representation.

Answer 2

If $X(t)X'(t)=X'(t)X(t)$, for all $t$, then $$ \frac{d}{dt}\mathrm{e}^{X(t)}=\mathrm{e}^{X(t)}\frac{dX(t)}{dt}. $$ If not, then the above does not hold.