For a differential equation of this form that has complex eigenvalues, can you choose either eigenvector to set up the general solution $$x(t) = e^{pt}\,S\,(\text{rotation matrix})\,S^{-1}\,x(0)\; ?$$ I got a different answer when I set up my $S$ matrix using the negative complex conjugate eigenvalue's eigenvector. I think it changed the spiral trajectory from going counterclockwise to clockwise?
I'm just confused because our teacher said you only need to use one of the eigenvectors to set up the solution, but it seems like you get different answers depending on which one you use.
This is what my textbook states:
