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Is there a simple way to solve the system of differential equations $$ \mathbf{P}'(t) = \mathbf{G} \mathbf{P}(t), $$

where $\mathbf{P}(t) = (p_{ij}(t))_{i,j \in \{1,2,\ldots, n\}}$ is an $n \times n$ matrix of functions and $\mathbf{G}$ is an $n \times n$ matrix of (real) constants? Of course, some extra hypothesis might be required of $\mathbf{G}$ (e.g. distinct eigenvalues) in order for there to be a simple (or even general) solution. Just let me know if that's the case.

I've only seen systems of the form

$$ \mathbf{x}' = \mathbf{G} \mathbf{x}, $$

where now $\mathbf{x}$ is a $n\times 1$ vector instead of an $n \times n$ matrix.

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up vote 2 down vote accepted

Your case is not more difficult than the case $x'=Gx$ since the different columns do not interfere. Just apply the theory for $x'=Gx$ column-wise.

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Thanks for the answer. Do you by chance know how to solve such a system using Mathematica? – Quinn Culver Dec 7 '11 at 21:51
@Quinn: I do. What system did you have in mind, so that we can work with a concrete example? – J. M. Dec 7 '11 at 22:54

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