Generic movement of eigenvalues for equivariant self-adjoint matrices
type: Item
instance of: scholarly article
title: Generic movement of eigenvalues for equivariant self-adjoint matrices
zbMATH Open document ID: 0827.65057
DOI: 10.1016/0377-0427(94)90032-9
author: Michael Dellnitz
author: Ian Melbourne
published in: Journal of Computational and Applied Mathematics
publication date: 1995-12-11T00:00:00Z
review text: In the numerical treatment of bifurcation problems one of the main tasks is to monitor the spectrum of matrices in a parametrized family. If the original problem possesses a geometrical symmetry with respect to a group of transformations, and if the discretization of the problem also respects this symmetry, then the Jacobian matrices arising in the linearization of the problem are also equivariant, i.e., \(\gamma A_\lambda = A_{\lambda \gamma}\) for all \(\gamma \in \Gamma\) where \(\Gamma \subset \text{GL} (n)\) is a compact Lie group. Let \(M^n_\Gamma\) denote the subset of self-adjoint \(n \times n\) real matrices which are equivariant with respect to \(\Gamma\). A subspace \(V\) of \(\mathbb{R}^n\) is \(\Gamma\) invariant if \(\gamma V \subset V\) holds for all \(\gamma \in \Gamma\). For a \(\Gamma\) invariant subspace, there is a unique decomposition into isotypic components \(V = W_1 \otimes \cdots \otimes W_k\) such that the subspaces \(W_j\) are invariant for each linear map \(A : V \to V\) which is equivariant. If the isotypic decomposition of an eigenspace of a matrix \(B \in M^n_\Gamma\) contains more than one irreducible subspace, this is called a \(\Gamma\) resonance for \(B\). A sample result of the paper is: Suppose the family \(A_\lambda \subset M^n_\Gamma\) undergoes a \(\Gamma\) resonance for \(\lambda = 0\) and \(E\) is the corresponding eigenspace. Then generically, \(E = U_1 \otimes U_2\), where \(U_1\) and \(U_2\) are \(\Gamma\) irreducible. The authors then discuss the cases when \(U_1\) and \(U_2\) are and are not isomorphic.
Mathematics Subject Classification ID: 65H17
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 15A18
zbMATH DE Number: 758547
zbMATH Keywords: eigenvalues
zbMATH Keywords: equivariant self-adjoint matrices
zbMATH Keywords: bifurcation
zbMATH Keywords: Jacobian matrices
zbMATH Keywords: compact Lie group
zbMATH Keywords: invariant subspace
zbMATH Keywords: isotypic components
zbMATH Keywords: isotypic decomposition
zbMATH Keywords: eigenspace
zbMATH Keywords: irreducible subspace
reviewed by: Eugene L. Allgower
MaRDI profile type: Publication
full work available at URL: 0377-0427(94)90032-9
OpenAlex ID: W2041338864
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description: scientific article; zbMATH DE number 758547