Eigenvalue & Bild
When images on a computer screen are flipped, rotated, or rescaled, it's the result of applying a matrix transformation to the image. Every pixel in the image is a vector in the xy-plane, and that pixel/vector is multiplied by a matrix to move it to its new location. Below, you can manipulate the entries of the transformation matrix using the sliders and see the effect on the smiley face. The eigenvalues of the matrix give information about the transformation as well; sometimes these are complex numbers, in which case they disappear.
Can you make the eigenvalues both positive? One positive and one negative? One eigenvalue equal to zero? Both complex? What does the transformed smiley face look like in each case?
More info on eigenvalues: http://mathworld.wolfram.com/Eigenvalue.html
Smiley face image: http://commons.wikimedia.org/wiki/File:718smiley.png