text: For dynamical systems, eigenvalues and eigenvectors tell you the long term, equilibrium behavior.  For a spacecraft, they tell you the modes of vibration of the spacecraft structure so that you can make sure that the accoustic environment of your launch vehicle won't shake it apart.\n\nMany, many examples.\n\nGeometrically, they show you special directions (invariant subspaces) of your vectors space that are simply stretched or shrunk, the direction remains the same.  Used for the projections that you see in advertising where it looks like a 3 dim figure is rotating.