Eigenvalues and eigenvectors in physics frequently characterize kind of free oscillations of a certain physical object. I mechanics it can be a membrane, in electromagnetics it can be e.g. a resonance cavity
In the last example eigenvalues mean the resonance frequencies of various modes. The corresponding eigenvectors decribe electromagnetic field distributions for these modes. Electromagnetic fileds
are determined with accuracy to a multipicative constant. This means that magnitudes of these field can not be determined.