Well, this is a tough one...anyway...
Consider a matrix, say, a #2xx2# square matrix #A#.
You can think at this matrix #A# as a magic wand...it operates its magic on vectors (represented by another matrix in form of a column).
So if you multiply #A# times your vector #v# as a consequence your vector magically changes and, for example, changes orientation (ok, you can do a lot of things with your magic matrix wand, but let us stay on easy ground)!!!
Now, imagine that there are some special vectors, called Eigenvectors, #v#, that when touched by your magic matrix do not change..well, they are only SCALED (they get bigger), they get stretched...only this!
How do you know of how much they were stretched? You have a scaling factor, the Eigenvalue , #lambda#, that tells you the amount of scaling of your special vector produced by your magic #A#.
In mathematical terms: #A*v=lambda*v# (=the action of #A# on #v# is to SCALE of an amount #lambda# the same #v#).
Ok...confusing...
Let see an example:
Consider a #2xx2# matrix #A#:
So if you multiply #A*v# you should get a stretching of 2 of your #v#:
(hope it is not too confusing!!!)
