Today i just created one application to demonstrate the use of eigen vectors.
It is only for 2 diamensional things. Beacuse in 3D calculating eigen vector is a complex process ( It depends), involving linear equation solving , and cubic equations' root finding.
You canse see the eigen axis can be used for finding the OBB( oriented bounding boxes) for a set of points. This has significance inmportant in Computer graphics.
One more interesting thing is eigen vector of a rotaion matrix is actually its axis of rotation (similar to curl of a vector field , but curl calculation is easier).