The

equation A

**v** = λ

**v** 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 http://www.physlink.com/Education/AskExperts/ae520.cfm). 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).