TITLE: Iterative method for matrix differential equation
QUESTION [6 upvotes]: Let $A$ and $X(t)$ be $n\times n$ matrices. I want to solve the matrix differential equation $$\dfrac{dX}{dt}(t)=AX(t)$$ with $X(0)=I$ (the $n\times n$ identity matrix) using the Picard iterative process.
So I want $X_0(t)=I$ for all $t\in\mathbb{R}$, and $$X_k(t)=I+\int_0^tAX_{k-1}(s)ds$$
So I compute $X_1(t)=I+\int_0^tAds=I+tA$.
Iterating, I have $X_2(t)=I+\int_0^tA(I+sA)ds=I+At+\dfrac{A^2t^2}{2}$.
Iterating once more, I have $X_3(t)=I+\int_0^tA(I+As+\dfrac{A^2s^2}{2})ds=I+At+\dfrac{A^2t^2}{2}+\dfrac{A^3t^3}{6}$
Then the limit will eventually be $X_\infty(t)=I+\sum_{k=1}^\infty\dfrac{(At)^k}{k!}$.
And it seems to be a solution of the original equation. How do I know that the iterative method is guaranteed to work in a case like this though? Is there some general theorem to verify that?

REPLY [2 votes]: The solution to your differential equation is the matrix exponential 
$$X(t)=e^{At} = \sum_{k=0}^\infty \frac{(At)^k}{k!}$$
which should always converge for any $A$ (Matrix exponential convergence).