On the equivariance properties of self-adjoint matrices

M. Dellnitz, B. Gebken, R. Gerlach, S. Klus, Dynamical Systems 35 (2020) 197–215.

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Journal Article | Published | English
Author
Dellnitz, Michael; Gebken, BennetLibreCat; Gerlach, RaphaelLibreCat; Klus, Stefan
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 ⊗nk=1Z2. 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)≃∏ki=1O(mi) 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.
Publishing Year
Journal Title
Dynamical Systems
Volume
35
Issue
2
Page
197-215
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Dellnitz M, Gebken B, Gerlach R, Klus S. On the equivariance properties of self-adjoint matrices. Dynamical Systems. 2020;35(2):197-215. doi:10.1080/14689367.2019.1661355
Dellnitz, M., Gebken, B., Gerlach, R., & Klus, S. (2020). On the equivariance properties of self-adjoint matrices. Dynamical Systems, 35(2), 197–215. https://doi.org/10.1080/14689367.2019.1661355
@article{Dellnitz_Gebken_Gerlach_Klus_2020, title={On the equivariance properties of self-adjoint matrices}, volume={35}, DOI={10.1080/14689367.2019.1661355}, number={2}, journal={Dynamical Systems}, author={Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan}, year={2020}, pages={197–215} }
Dellnitz, Michael, Bennet Gebken, Raphael Gerlach, and Stefan Klus. “On the Equivariance Properties of Self-Adjoint Matrices.” Dynamical Systems 35, no. 2 (2020): 197–215. https://doi.org/10.1080/14689367.2019.1661355.
M. Dellnitz, B. Gebken, R. Gerlach, and S. Klus, “On the equivariance properties of self-adjoint matrices,” Dynamical Systems, vol. 35, no. 2, pp. 197–215, 2020, doi: 10.1080/14689367.2019.1661355.
Dellnitz, Michael, et al. “On the Equivariance Properties of Self-Adjoint Matrices.” Dynamical Systems, vol. 35, no. 2, 2020, pp. 197–215, doi:10.1080/14689367.2019.1661355.

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