The best Euclidean fit to a given distance matrix in prescribed dimensions
type: Item
instance of: scholarly article
title: The best Euclidean fit to a given distance matrix in prescribed dimensions
zbMATH Open document ID: 0569.15015
DOI: 10.1016/0024-3795(85)90181-8
author: R. Mathar
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137073
review text: A distance matrix is defined to be a real symmetric matrix with zero diagonal and non-positive off-diagonal entries. An \(n\times n\) distance matrix \([d_{i_ j}]\) is said to be Euclidean if there exists an integer \(k\leq n\) and points \(x_ 1,...,x_ n\in {\mathbb{R}}^ k\) such that \(d_{i_ j}=-(1/2)\| x_ i-x_ j\|^ 2\), where \(\|.\|\) is the Euclidean norm on \({\mathbb{R}}^ k\). Given a distance matrix D and a fixed \(k<n\) the author finds the closest Euclidean distance matrix to D with respect to a wide class of norms on the space of symmetric \(n\times n\) matrices. The construction uses diagonalization. It is motivated by problems of scaling and clustering of multidimensional data.
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 62H30
Mathematics Subject Classification ID: 65D10
zbMATH DE Number: 3908599
zbMATH Keywords: Euclidean distance matrix
zbMATH Keywords: diagonalization
zbMATH Keywords: scaling
zbMATH Keywords: clustering
zbMATH Keywords: multidimensional data
MaRDI profile type: Publication
OpenAlex ID: W2039665922
cites work: Extreme properties of eigenvalues of a hermitian transformation and singular values of the sum and product of linear transformations
cites work: Q5800822
cites work: Q3954439
cites work: Some properties of clasical multi-dimesional scaling
cites work: Inequalities: theory of majorization and its applications
cites work: Remarks to Maurice Frechet's article ``Sur la definition axiomatique d'une classe d'espaces vectoriels distancies applicables vectoriellement sur l'espace de Hilbert''
cites work: An Extremum Property of Sums of Eigenvalues
description: scientific article; zbMATH DE number 3908599