Approximation by matrices positive semidefinite on a subspace
type: Item
instance of: scholarly article
title: Approximation by matrices positive semidefinite on a subspace
zbMATH Open document ID: 0667.15020
DOI: 10.1016/0024-3795(88)90202-9
published in: Linear Algebra and its Applications
publication date: 1988-01-01T00:00:00Z
review text: The authors study the following problem: Let \(A\in {\mathbb{R}}^{n\times n}\), \(S\in {\mathbb{R}}^{n\times p}\) \((p<n)\), rank S\(=p\), minimize \(\| A- P\|\), subject to \(P=P^ T\), \(x^ TPx\geq 0\) \(\forall x:\) \(S^ Tx=0\). Here \(\| A\|\) is either the Frobenius norm \((\| A\| =(\sum | a_{ij}|^ 2)^{1/2})\) or the 2-norm \((\| A\|_ 2=\sup (\| Ax\| | \quad \| x\| =1),\quad \| x\| =(\sum | x_ i|^ 2)^{1/2}).\) The authors describe the (unique) minimizing matrix in the case of the Frobenius norm and a minimizing matrix in the case of the 2-norm when A is symmetric. The results are applied to give two new characterizations for Euclidean distance matrices.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 4091674
zbMATH Keywords: approximation of matrices
zbMATH Keywords: Frobenius norm
zbMATH Keywords: 2-norm
zbMATH Keywords: minimizing matrix
zbMATH Keywords: Euclidean distance matrices
author: Thomas L. Hayden
author: James H. Wells
MaRDI profile type: Publication
full work available at URL: 0024-3795(88)90202-9
OpenAlex ID: W1989148040
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description: scientific article; zbMATH DE number 4091674