1. The problem statement, all variables and given/known data This isn't really a question in particular. I am doing my first Differential Equations course, and in the complex eigenvalues part, I am getting confused as to how to find the eigenvectors. Example: Solve for the general solution of: x' = (1 -1)x (don't know how to type a matrix using latex sorry) (5 -3) 2. Relevant equations I know how to find the eigenvectors if there were real eigenvalues, since I've taken Linear Algebra and know that you can just simply reduce the matrix into Gauss-Jordan form. 3. The attempt at a solution The eigenvalues are -1 +/- 2i (this is the easy part) What confuses me is the next step: The examples usually only plug in one eigenvalue (say -1+2i), which I don't know why. And so the matrix will now look something like this; (2-2i -1) ( 5 -2-2i) What happens next I don't understand, usually the example would take (2-2i)v1 - v2 = 0 if (v1,v2) was one of the eigenvectors And it is around here I get the most confused. - do I use v1 or v2 as the free variable? - do I use the top or bottom row (5v1 + (-2-2i)v2 = 0) to find v1 and v2? I have tried reading the books and the examples but they never show what they exactly do.