Based on Wikipedia's entry on eigen vectors, Av=kv means the matrix A transformation will preserve the directionality along eigen vectors v. The largest eigen value indicates the strongest transformation along that eigen vector.
The transformation matrix ![\bigl[ \begin{smallmatrix} 2 & 1\\ 1 & 2 \end{smallmatrix} \bigr]](http://upload.wikimedia.org/math/0/7/7/077bd41a8b65597290e3cb5cd869a74e.png)
preserves the direction of vectors parallel to 
(in blue) and 
(in violet). The points that lie on the line through the origin, parallel to an eigenvector, remain on the line after the transformation. The vectors in red are not eigenvectors, therefore their direction is altered by the transformation. Notice that the blue vectors are scaled by a factor of 3. This is their associated eigenvalue. The violet vectors are not scaled, so their eigenvalue is 1.
Source: http://upload.wikimedia.org/wikipedia/commons/0/06/Eigenvectors.gif
A Youtube demo is also helpful: http://www.youtube.com/watch?v=wXCRcnbCsJA
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