Hermitian matrices depending on three parameters: Coalescing eigenvalues
type: Item
instance of: scholarly article
title: Hermitian matrices depending on three parameters: Coalescing eigenvalues
zbMATH Open document ID: 1244.15005
author: Luca Dieci
author: Alessandro Pugliese
published in: Linear Algebra and its Applications
publication date: 2012-05-11T00:00:00Z
review text: The behavior of eigenvalues under perturbation is a classical problem. Eigenvalues of a matrix valued functions can always be chosen as continuous functions of the parameters, but further smoothness is generally precluded multiple eigenvalues are encountered. However, severe lack of smoothness can be expected whenever there are multiple eigenvalues. The authors study what happens when one or more eigenvalues of a matrix depending on parameters come together (coalescing eigenvalues) for Hermitian matrix valued functions depending on three parameters that vary in a bounded surface of \({\mathbb R}^3\). They are interested in giving global, hence topological, localization results for values in parameter space where the eigenvalues coalesce, and their particular emphasis is on Hermitian matrices depending on three parameters. Some pictures are helpful on idea illustration.
reviewed by: Tin-Yau Tam
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 6033310
zbMATH Keywords: Hermitian matrix valued functions
zbMATH Keywords: minimum variation
zbMATH Keywords: phase rotating/preserving
zbMATH Keywords: Berry phase
zbMATH Keywords: multiple eigenvalues
zbMATH Keywords: coalescing eigenvalues
MaRDI profile type: Publication
full work available at URL: j.laa.2012.01.009
OpenAlex ID: W2046602128
cites work: Quantal phase factors accompanying adiabatic changes
cites work: Numerical computation of an analytic singular value decomposition of a matrix valued function
cites work: Smoothness and periodicity of some matrix decompositions
cites work: On Smooth Decompositions of Matrices
cites work: Locating coalescing singular values of large two-parameter matrices
cites work: Two-Parameter SVD: Coalescing Singular Values and Periodicity
cites work: A proof of the Smale conjecture, \(Diff(S^ 3)\simeq O(4)\)
cites work: A global analysis of matrices of functions of several variables
cites work: Q3955947
cites work: Q5443078
cites work: Q4382059
cites work: Q4457537
cites work: Some global properties of matrices of functions of one variable
cites work: Singularities in the spectra of random matrices
DOI: 10.1016/J.LAA.2012.01.009
description: scientific article; zbMATH DE number 6033310