How to use matrices to solve systems of ODEs?

[medstudent]

I'm dealing with systems of 3 differential equations that are all coupled to each other. Fortunately, all the ODEs are first order.

Can somebody give me a primer of how to use matrices to solve these problems?

here's an example:

Say we have a system of 3 ODEs all coupled to each other:

Mx, My, Mz

dMx/dt = A*Mx + B*My + C*Mz

dMy/dt = D*Mx + E*My + F*Mz

dMz/dt = G*Mx + H*My + J*Mz


So the matrix would be:

dM/dt = [A B C
D E F
G H J] M + [K L M]

So what do I do now? Diagonalize the matrix and then find eigenvectors? Whats the step by step?

[arildno]

That's right.
Now, suppose you've got a diagonal matrix.
What are the functions satisfying those differential equations?