On the equivariance properties of self-adjoint matrices

Michael Dellnitz; Bennet Gebken; Raphael Gerlach; Stefan Klus

doi:10.1080/14689367.2019.1661355

On the equivariance properties of self-adjoint matrices

Michael Dellnitz, Bennet Gebken, Raphael Gerlach, Stefan Klus^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

Abstract

We investigate self-adjoint matrices A ∈ R^n,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ₂(A) ⊂ O(n) which is isomorphic to (Formula presented.). If the self-adjoint matrix possesses multiple eigenvalues–this may, for instance, be induced by symmetry properties of an underlying dynamical system–then A is even equivariant with respect to the action of a group (Formula presented.) where m₁, . . . ,m_k are the multiplicities of the eigenvalues λ₁, . . . , λ_k of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.

Original language	English
Pages (from-to)	197-215
Number of pages	19
Journal	Dynamical Systems
Volume	35
Issue number	2
DOIs	https://doi.org/10.1080/14689367.2019.1661355
Publication status	Published - 2 Apr 2020

Keywords

15A24
15B57
37G40
41A58
bifurcation theory
equivariance
Procrustes problem
Self-adjoint matrix
Taylor expansion

ASJC Scopus subject areas

General Mathematics
Computer Science Applications

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10.1080/14689367.2019.1661355

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title = "On the equivariance properties of self-adjoint matrices",

abstract = "We investigate self-adjoint matrices A ∈ Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A) ⊂ O(n) which is isomorphic to (Formula presented.). If the self-adjoint matrix possesses multiple eigenvalues–this may, for instance, be induced by symmetry properties of an underlying dynamical system–then A is even equivariant with respect to the action of a group (Formula presented.) where m1, . . . ,mk are the multiplicities of the eigenvalues λ1, . . . , λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.",

keywords = "15A24, 15B57, 37G40, 41A58, bifurcation theory, equivariance, Procrustes problem, Self-adjoint matrix, Taylor expansion",

author = "Michael Dellnitz and Bennet Gebken and Raphael Gerlach and Stefan Klus",

note = "Funding Information: This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures of the Deutsche Forschungsgemeinschaft. We also thank an anonymous reviewer for important comments on the contents of this article. Publisher Copyright: {\textcopyright} 2019, {\textcopyright} 2019 Informa UK Limited, trading as Taylor & Francis Group.",

year = "2020",

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doi = "10.1080/14689367.2019.1661355",

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AU - Gebken, Bennet

AU - Gerlach, Raphael

AU - Klus, Stefan

N1 - Funding Information: This work is supported by the Priority Programme SPP 1881 Turbulent Superstructures of the Deutsche Forschungsgemeinschaft. We also thank an anonymous reviewer for important comments on the contents of this article. Publisher Copyright: © 2019, © 2019 Informa UK Limited, trading as Taylor & Francis Group.

PY - 2020/4/2

Y1 - 2020/4/2

N2 - We investigate self-adjoint matrices A ∈ Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A) ⊂ O(n) which is isomorphic to (Formula presented.). If the self-adjoint matrix possesses multiple eigenvalues–this may, for instance, be induced by symmetry properties of an underlying dynamical system–then A is even equivariant with respect to the action of a group (Formula presented.) where m1, . . . ,mk are the multiplicities of the eigenvalues λ1, . . . , λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.

AB - We investigate self-adjoint matrices A ∈ Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A) ⊂ O(n) which is isomorphic to (Formula presented.). If the self-adjoint matrix possesses multiple eigenvalues–this may, for instance, be induced by symmetry properties of an underlying dynamical system–then A is even equivariant with respect to the action of a group (Formula presented.) where m1, . . . ,mk are the multiplicities of the eigenvalues λ1, . . . , λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.

KW - 15A24

KW - 15B57

KW - 37G40

KW - 41A58

KW - bifurcation theory

KW - equivariance

KW - Procrustes problem

KW - Self-adjoint matrix

KW - Taylor expansion

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DO - 10.1080/14689367.2019.1661355

M3 - Article

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SN - 1468-9367

VL - 35

SP - 197

EP - 215

JO - Dynamical Systems

JF - Dynamical Systems

IS - 2

ER -

On the equivariance properties of self-adjoint matrices

Abstract

Keywords

ASJC Scopus subject areas

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