Given a finite metric space $(X = \{ x_i \}_{i=1}^n,d)$, one can form the matrix $A$ of pairwise distances $a_{ij} = d(x_i, x_j)$.
What does the eigenspectrum of this matrix say about the metric $d$? Considering the success of spectral methods for analyzing various matrices formed from graphs, this seems like a natural thing to do. If it is a thing, what is it called and where should I look for more information?