On the equivariance properties of self-adjoint matrices. Issue 2 (2nd April 2020)

• Record Type: Journal Article
• Authors: Dellnitz, Michael; Gebken, Bennet; Gerlach, Raphael; Klus, Stefan
• Is Part Of: Dynamical systems. Volume 35:Issue 2(2020)
• Page Start: 197
• Page End: 215
• Publication Date: 2020-04-02
• Subjects: Self-adjoint matrix -- equivariance -- bifurcation theory -- Procrustes problem -- Taylor expansion; 15B57 -- 15A24 -- 37G40 -- 41A58; Differentiable dynamical systems -- Periodicals; 515.35205
• DOI: 10.1080/14689367.2019.1661355
• Languages: English
• ISSNs: 1468-9367

Abstract

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 Γ 2 ( A ) ⊂ O ( n ) which is isomorphic to ⊗ k = 1 n Z 2 . 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 Γ ( A ) ≃ ∏ i = 1 k O ( m i ) where m 1, …, m k are the multiplicities of the eigenvalues λ 1, …, 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.