In some physics problems it is sometimes useful to define a distance matrix for a system of particles with positions denoted by $x_1$, ..., $x_N$. Then the matrix would be given by $$M_{ij}=f(x_i-x_j)+\alpha\delta_{ij}\sum_{k}_{k\ne i}f(x_i-x_k)$$ where $f$ is a function yet to be defined and $\alpha$ is a constant parameter, which can be specified for different models.
I know that the matrix $M$ contains information about the position correlations of the $N$ particles. But what do the eigenvalues represent and what are the physical meaning for the eigenvectors?