The equation Av = λv can be interpreted so that eigenvectors and eigenvalues can be thought of in a somewhat more intuitive fashion than simply their definitions in this equation. This is good news for physicists. We can express this equation in words by saying the following: take a vector, and transform it; if the new, transformed vector is simply a multiple of the old one, then that vector is an eigenvector. The multiplier is an eigenvalue. (N.B. "Eigenvector" literally means "own vector" (correct me if I'm wrong) in German, and we can now see why they are so called: the transformation of a vector creates a multiple of itself).

A nice way to visualise such a transformation is that of a rubber square with an arrow drawn on it (see If you stretch the square along a particular axis, only arrows in certain directions will keep their direction; this is true no matter how hard you stretch the beast. We can then say that those directions (vectors) are eigenvectors for that transformation (stretching), and the eigenvalues (length of the new arrow compared to the old arrow) depend on how hard you stretch (which is inherent in the transformation matrix).