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I'm trying to work out how to find $$\exp(At)$$ for a system of linear differential equations $$x'=Ax.$$

I know that the solution is a fundamental matrix of the system such that $$\exp(At)=I$$ at time $0$.

What is the method for solving this using the Laplace transform? The only method I can figure out is finding the eigenvalues and diagonalzing the matrix.

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Look at "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later", Moler & Van Loan –  copper.hat Jul 31 '12 at 22:01

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The Laplace transform of $exp(At)$ is $(sI-A)^{-1}$. So compute that latter and then take its inverse Laplace transform.

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How do you propose to compute $(sI-A)^{-1}$? –  copper.hat Aug 1 '12 at 0:18
    
Yes. Then take element wise inverse Laplace transform. What we will have is precisely $exp(At)$. –  Manos Aug 1 '12 at 12:25
    
Computing that inverse seems to be a nightmare. You cannot use normal elimination, so how do you compute that inverse in actual problems? –  rmh52 Aug 1 '12 at 16:03
    
@Manos: I understand the definition and the formal computation, but practicality is another matter. Just as one almost never inverts a matrix using Cramer's rule. –  copper.hat Aug 1 '12 at 16:35
    
Everything depends on what our matrix $A$ is. If $A$ is $3 \times 3$, computing $(sI-A)^{-1}$ is easy. If it is $1000 \times 1000$ it is not. In any case, there is a variety of ways to compute the matrix exponential. The above mentioned article is a good guide. –  Manos Aug 1 '12 at 22:43

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