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We study asymptotics of the eigenvalues and eigenfunctions of the operatorsused for constructing multidimensional scaling (MDS) on compact connectedRiemannian manifolds, in particular on closed connected symmetric spaces. Theyare the limits of eigenvalues and eigenvectors of squared distance matrices ofan increasing sequence of finite subsets covering the space densely in thelimit. We show that for products of spheres and real projective spaces, thenumbers of positive and negative eigenvalues of these operators are bothinfinite. We also find a class of spaces (namely $\mathbb{RP}^n$ with odd$n>1$) whose MDS defining operators are not trace class, and original distancescannot be reconstructed from the eigenvalues and eigenfunctions of theseoperators.

On eigenvalues and eigenfunctions of the operators defining multidimensional scaling on some symmetric spaces