How I solve a matrix-based ODE problem?

Question

I'm very confused about solving an ODE problem with matrices. I have the following equation:

d(Phi)/dt = A * Phi

Where Phi and A are both matrices. I also have an initial condition matrix Phi(t0). I'm really struggling with how to get Phi from this. If it were just numbers, it's a simple seperable ODE. But matrices were always my weak point. Can anyone help??

Answer 1

Ok,
it is still a simple separable ODE - except it's just in matrix form.
I'm thinking that Phi is a column vector and A is an nxn matrix of constants, yes?

So, the solution to your ODE is Phi(t) = exp(At)*Phi(t0), where exp(At) is the matrix exponential.

exp(At) = (At)^0/0! + (At)^1/1! + (At)^2/2! + ...

That's about all I can recall from my Graduate ODE courses!

Answer 2

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