Eigenanalysis and metric multidimensional scaling on hierarchical structures

The known hierarchical clustering scheme is equivalent to the concept of ultrametric distance. Every distance can be represented in a spatial model using multidimensional scaling. We relate both classes of representations of proximity data in an algebraic way, obtaining some results and relations on clusters and the eigenvalues of the inner product matrix for an ultrametric distance. Principal coordinate analysis on an ultrametric distance gives two classes of independent coordinates, describing compact clusters and representing objects inside every cluste