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!
 A: I will try to focus on 3 topics raised in your answer.  


*

*Convergence of exp(tM)


Let $M$ be an $n\times n$ matrix with coefficients in the field $\mathbb K$ (take $\mathbb R$ for example). The norm of $M$, denoted by $\| M\|$ is given by 
$\| M \|^2 := \sum_{i,j=1}^n M_{ij}^2$. You can prove that $\| MN\|\leq \| M\| \| N\|$ for any matrices $M$ and $N$ as above using the Cauchy-Schwarz inequality.
The exponential
$\exp(tM):=\sum_{i= 0}^{\infty} \frac{(tM)^i}{i!}$
converges absolutely for all $M$; the proof works by induction.


*

*Multiplication of exponential matrices


The nice formula
$\exp(M)\exp(N)=\exp(M+N)$
holds if $MN-NM=0$; in this sense the exponential map for matrices is not a straightforward generalization of the one for real numbers. You can prove the above relation by using the definition of $\exp(M)$ and $\exp(N)$: rearranging indices is allowed due to absolute convergence. If $MN-NM=0$ does not hold, then a very interesting but rather complicated formula for the product $\exp(M)\exp(N)$ holds: it is called the Campbell-Baker-Hausdorff formula.


*

*On the derivative of $\exp(tM)$.


Use the definition 
$\frac{d\exp(tM)}{dt}=\lim_{h\rightarrow 0}\frac{\exp((t+h)M)-\exp(tM)}{h}=(tMhM=hMtM!)= \lim_{h\rightarrow 0}\exp(tM)\frac{\exp(hM)-1}{h}=\exp(tM)
\lim_{h\rightarrow 0}\frac{\exp(hM)-1}{h}$;
Using the definition of $\exp(hM)=1+hM+O(h^2)$, you can write 
$\frac{\exp(hM)-1}{h}=M+O(h^1)$,
arriving at 
$\frac{d\exp(tM)}{dt}=\exp(tM)M=M\exp(tM)$,
as $M$ and $\exp(tM)$ commutes.
A: 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 $A$.
