Sensitivity analysis of all eigenvalues of a symmetric matrix
type: Item
instance of: scholarly article
title: Sensitivity analysis of all eigenvalues of a symmetric matrix
zbMATH Open document ID: 0816.15016
published in: Numerische Mathematik
publication date: 1995-07-23T00:00:00Z
review text: Let \(A = A(x)\) be a symmetric \(n \times n\) matrix whose elements depend smoothly on the \(p\)-vector \(x\). This paper studies the sensitivity of the eigenvalues \(\lambda_ k (x) = \lambda_ k (A(x))\) as a function of \(x\). Because \(\lambda_ k (x)\) is not convex, the functions \(\sigma_ m(x) = \sum_{k=1}^ m \lambda_ k (x)\) are introduced which turn out to be convex. Since moreover \(\lambda_ k (x)\) and \(\sigma_ m(x)\) are nonsmooth, the derivations depend heavily on results from nonsmooth and convex analysis. Another important element in the analysis is the separation of the effects of the maps \(A \mapsto \lambda_ m(A)\) and of \(x \mapsto A(x)\). First \(\sigma_ m\) is characterized by Ky Fan's variational principle, i.e., \(\sigma_ m = \max \{\text{tr} (A XX^ T)\}\) where the maximum is taken over all \(n \times m\) matrices \(X\) satisfying \(X^ TX = I_ m\). Next the subdifferential calculus for the map \(A \mapsto \lambda_ m (A)\) is analysed. Finally the generalized gradient and the directional derivatives for \(\lambda_ m(x)\) are explicitly computed.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 15A42
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 52A41
zbMATH DE Number: 739396
zbMATH Keywords: symmetric matrix
zbMATH Keywords: sensitivity of eigenvalues
zbMATH Keywords: nonsmooth analysis
zbMATH Keywords: convex functions
zbMATH Keywords: subdifferential calculus
zbMATH Keywords: generalized gradient
zbMATH Keywords: directional derivatives
author: Jean-Baptiste Hiriart-Urruty
author: Dongyi Ye
MaRDI profile type: Publication
full work available at URL: s002110050109
OpenAlex ID: W2016624757
DOI: 10.1007/S002110050109
description: scientific article; zbMATH DE number 739396