Optimal ellipsoids and decomposition of positive definite matrices
type: Item
instance of: scholarly article
title: Optimal ellipsoids and decomposition of positive definite matrices
zbMATH Open document ID: 1117.15024
author: N. E. Zubov
published in: Journal of Mathematical Analysis and Applications
publication date: 2007-04-26T00:00:00Z
review text: Let \(A\) be a positive definite matrix. The aim of the paper is to give a characterization of the unique diagonal matrix \(D\) with \(D\geq A\) with the smallest determinant. Such a matrix \(D\) is the optimizer if and only if (a) \(D\geq A\) and \(D-A\) is singular, and (b) there is an \(n\times n\) matrix \(X\geq 0\) with \((D-A)X=0\) and \(\text{diag} (X)=D^{-1}\). This problem has the following geometrical interpretation. Given an ellipsoid \(\mathcal A\) in \(\mathbb R^n\), find the ellipsoid \(\mathcal D\), contained in \(\mathcal A\), with principal axes parallel to the coordinate axes, of the largest volume. Some other consequences of this optimization problem are also derived.
reviewed by: Andreas Arvanitoyeorgos
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 5146751
zbMATH Keywords: ellipsoid
zbMATH Keywords: determinant maximazation
zbMATH Keywords: John's theorem
zbMATH Keywords: positive definite matrix
MaRDI profile type: Publication
full work available at URL: j.jmaa.2006.09.065
OpenAlex ID: W2009738439
cites work: Q4387224
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cites work: Determinant Maximization with Linear Matrix Inequality Constraints
DOI: 10.1016/J.JMAA.2006.09.065
description: scientific article; zbMATH DE number 5146751