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 journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)197-215
Number of pages19
JournalDynamical Systems
Issue number2
Publication statusPublished - 2 Apr 2020


  • 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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