# Solving a linear system of ODEs given by a matrix product

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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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.