On certain linear mappings between inner-product and squared-distance matrices
type: Item
instance of: scholarly article
title: On certain linear mappings between inner-product and squared-distance matrices
zbMATH Open document ID: 0644.15003
DOI: 10.1016/0024-3795(88)90006-7
author: Frank Critchley
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15A12
zbMATH DE Number: 4049782
zbMATH Keywords: inner product matrices
zbMATH Keywords: squared distance matrices
zbMATH Keywords: inner product space
zbMATH Keywords: Spectral decompositions
zbMATH Keywords: inverse
zbMATH Keywords: adjoint mappings
zbMATH Keywords: multidimensional scaling
reviewed by: Hans Havlicek
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90006-7
OpenAlex ID: W2147902861
cites work: Q3942241
cites work: Q3793551
cites work: Q4749029
cites work: Q3954439
cites work: Properties of Euclidean and non-Euclidean distance matrices
cites work: The relation between hierarchical and Euclidean models for psychological distances
cites work: Some boundary conditions for a monotone analysis of symmetric matrices
cites work: Q5566063
cites work: The best Euclidean fit to a given distance matrix in prescribed dimensions
cites work: Q3707170
cites work: Quadratic entropy and analysis of diversity
description: scientific article; zbMATH DE number 4049782