exponential matrix Hi i am trying to understand the exponential matrix:
When is exponential matrix function $e^{At}$ integrable where A is an $n \times n$ matrix and $t$ is an $n$-dimensional vector? By integrable i mean indefinite integral over $(0, \infty)$
Do we have to say something about the norm of the matrix?
Can we say the following if integral of $e^{At}$ is infinity $e^{-At}$ is integrable?
Could you give an example of a matrix $A$ where the integral $e^{At}$ does not converge to 
any real number or plus or minus infinity?
Could we say when $e^{At}$ is integrable then $e^{At} \dot e^{At}$ is integrable as well i.e. the dot product of these two same exponential matrices?
What is the explicit representation for $e^{At} \dot e^{At}$? Similarly how is the derivative of this object defined why is the derivative equal to $Ae^{At}$
Thanks a lot for your answer!

Hint: take a look at Jordan normal form of a matrix. It definitely helps with understanding how analytical functions work with matrices.
As an example, $e^{At}$ is integrable on $(0,\infty)$ $\iff$ $\forall j\, \Re(\lambda_j(A))<0$, where $\lambda_j(A)$ are the eigenvalues of $A$.
