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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 Loan2012-07-31

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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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    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.2012-08-01