## 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 (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 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.

Original language | English |
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Pages (from-to) | 197-215 |

Number of pages | 19 |

Journal | Dynamical Systems |

Volume | 35 |

Issue number | 2 |

DOIs | |

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