This is the question I got: We've seen that if the eigenvalues of a 2 by 2 matrix are purely imaginary, then the flow induced by the equation $\frac{dY}{dt} = AY$ acts by rotation.
A. What condition on the eigenvalues and eigenvectors can you check to determine whether the rotation is clockwise or counterclockwise?
B. Does this work for spirals, too?
Usually when I determine the direction of rotation, I just choose a point and plug into the equation to see the direction of the vector and inspect. But this question is asking me if there're some condition on the eigenvectors and eigenvalues to tell that. Since this question is specifically about purely imaginary eigenvalues case, i.e., $\lambda = \pm bi$, I don't have a clue about it.