A Nice Explanation of Eigen Values.

In this shear transformation of the Mona Lisa the picture was deformed in such a way that its central vertical axis was not modified. (Note: The corners have been cropped on the right hand picture.) The blue vector, from her chest to her shoulder, has changed direction, but the red one, from her chest to her chin, is unchanged. The red vector is thus an eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors along the same vertical line are also eigenvectors, with the same eigenvalue. They form the eigenspace for this eigenvalue.