This is my example but your welcome to elaborate if you have better examples. Suppose we have a system of differential equations
$$\frac{dR}{dt}=aJ \\ \frac{dJ}{dt}=bR$$
s.t. $a,b \gt 0$. I've determined that the eigenvalues are equal and are greater than zero by using the Jacobian $(A)$ and evaluating the determinant, $\det(A-r \mathbb{I})$, to get the characteristic equation. Thus we have an unstable improper or proper node. How would you determine if its an improper or a proper node from the given differential equations? I know it has to do something with having two independent eigenvectors or one independent eigenvector.