When images on a computer screen are flipped, rotated, or rescaled, it's the result of applying a [b]matrix transformation[/b] to the image. Every pixel in the image is a vector in the xy-plane, and that pixel/vector is multiplied by a [math]2 \times 2[/math] 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 [b]eigenvalues[/b] of the matrix give information about the transformation as well; sometimes these are complex numbers, in which case they disappear. [br][br]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? [br][br]More info on eigenvalues: [url]http://mathworld.wolfram.com/Eigenvalue.html[/url][br]Smiley face image: [url]http://commons.wikimedia.org/wiki/File:718smiley.png[/url]