In various math libraries developed for game engines, I see 2x2 and 3x3 square matrices having this function (along with some helper functions). I tried to find information regarding it but did not come across game specific material (wikipedia entry is quite heavy and does not relate to games).
It would be great if in your answer, you could provide an example or two on how it is used in games.
Eigenvalues and eigenvectors are very commonly used in math, so it's good to understand them even if you don't have an immediate use for the in games. They're generally covered near the end of an undergraduate-level linear algebra course.
One I can think of is for dealing with the moment of inertia in physics. This is a symmetric positive-definite matrix, so all the eigenvalues are guaranteed to be real and positive. If you compute the principal axes of a rigid body (which are the eigenvectors of the matrix), you can rotate the body so that its moment of inertia is diagonal - this would allow you to store just the diagonal components as a vector3, instead of the full matrix.
Another place I've seen them used is computing the best-fit plane to a set of points, or to find the best-fit bounding box (section 6.1.2 in the link)
Have you searched in the WildMagic or Ogre code to see where else they're used?
As far as I know, eigenvalue decomposition is a middle-step that can be performed in some algorithms for things with more readily apparent uses, such as matrix inverting.
