  We study asymptotics of the eigenvalues and eigenfunctions of the operators
used for constructing multidimensional scaling (MDS) on compact connected
Riemannian manifolds, in particular on closed connected symmetric spaces. They
are the limits of eigenvalues and eigenvectors of squared distance matrices of
an increasing sequence of finite subsets covering the space densely in the
limit. We show that for products of spheres and real projective spaces, the
numbers of positive and negative eigenvalues of these operators are both
infinite. 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 distances
cannot be reconstructed from the eigenvalues and eigenfunctions of these
operators.
