What is the importance of eigenvalues/eigenvectors? What is the importance of eigenvalues/eigenvectors?  

I think if you want a better answer, you need to tell us more precisely what you
  may have in mind: are you interested in theoretical aspects of eigenvalues; do
  you have a specific application in mind? Matrices by themselves are just arrays of
  numbers, which take meaning once you set up a context. Without the context, it seems
  difficult to give you a good answer. If you use matrices to describe adjacency relations,
  then eigenvalues/vectors may mean one thing; if you use them to represent linear maps
  something else, etc.
One possible application: In some cases, you may be able to diagonalize your
  matrix $M$ using the eigenvalues, which gives you a nice expression for $M^k$. 
  Specifically, you may be able to decompose your matrix into a product
  $SDS^{-1}$ , where $D$ is diagonal, with entries the eigenvalues, and $S$ is the
  matrix with the associated respective eigenvectors.
    I hope it is not a problem to post this as a comment. I got a couple of Courics
  here last time for posting a comment in the answer site.
Mr. Arturo:
   Interesting approach!. This seems to connect with the theory of characteristic
   curves in PDE's(who knows if it can be generalized to dimensions higher than 1), which
   are curves along which a PDE becomes an ODE, i.e., as you so brilliantly said,
   curves along which the PDE decouples. 
