On certain linear mappings between inner-product and squared-distance matrices
Let \(X=\{x_ i|\) \(i=1,...,n\}\) be a set of \(n>2\) points in a real inner product space such that \(\sum x_ i=0\), i.e. X is centered. Then there is a one-to-one correspondence between the matrices \(b_{ij}=<x_ i,x_ j>\) and \(d_{ij}=\| x_ i-x_ j\|^ 2\). The author shows that this can be extended to a linear isomorphism of the vector spaces spanned by all these matrices \((b_{ij})\) and \((d_{ij})\), respectively. Spectral decompositions for both this isomorphism and its inverse as well as their adjoint mappings are given. Applications to multidimensional scaling are discussed.
scientific article; zbMATH DE number 4049782
scientific article