Suppose that we have a $2\times 2$ linear mapping given by a matrix $A$. There are three cases: 1) the eigenvalues $\lambda_1\neq \lambda_2$ are real, 2) there is only one real eigenvalue $\lambda$, 3) there are two complex eigenvalues satisfying $\lambda_1 = \overline{\lambda_2}$.
In each case I would like to derive the exact equations of the curves that are invariant under this mapping. For instance, in the first case, when $1 > |\lambda_1| > |\lambda_2|$, these curves should be exponential functions, but how to derive the exact equations for them? And how to do the analogous for all other cases? I would just like to use elementary linear algebra here.
Thank you very much for some help.