| | Applications of Eigenvectors!?
Since math is your hobby--here is a pure MATHEMATICAL application.
Consider the second fundamental form of a surface z=f(x,y) in three space. This
is the matrix of second derivatives of z, when co-ordinates are chosen so that
surface is tangent at the origin and the normal vector is perpendicular to the
The product of the eigenvalues is the gaussian curvature of the surface and the
average is the mean curvature.
This theory extends to n-1 dimensional submanifolds of n space.
You can find this in any elementary book on differential geometry such as the
Schaum's outline on Differential geometry or the book by Frank Morgan.