In some instances, like physics, you may find that quantities you are after are eigenvalues of matrices. However, for example, explaining that "the mass of a muon is an eigenvalue of a matrix" does not really give much insight into where the value comes from.
I know the general definition of an eigenvalue is the amount something is transformed along the eigenvectors. But in terms of integer-valued symmetric matrices, is there any other definition, for example in terms of lattices, that gives a geometric description of an eigenvalue?
For example, one might envisage that a general symmetric integer valued matrix might transform a regular lattice, into another lattice which also had vertices at integer values. And the eigenvalues might be distances between particular vertices in the lattice.
Basically I'm looking for another equivalent definition to describe eigenvalues in terms of the integer values of its matrix.