# On the Equivariance Properties of Self-adjoint Matrices

## TL;DR

This paper characterizes the equivariance properties of self-adjoint matrices, linking their symmetry groups to eigenvalue multiplicities, with implications for bifurcation theory, graph symmetries, and matrix approximation problems.

## Contribution

It provides a complete characterization of the symmetry groups of self-adjoint matrices based on their eigenvalues and multiplicities, extending understanding of their equivariance properties.

## Key findings

- Self-adjoint matrices are equivariant under a specific group isomorphic to a product of Z2 groups.
- Multiple eigenvalues induce larger symmetry groups isomorphic to products of orthogonal groups.
- Results have applications in bifurcation analysis, graph symmetry, and matrix approximation problems.

## Abstract

We investigate self-adjoint matrices $A\in\mathbb{R}^{n,n}$ 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 $\Gamma_2(A)\subset \mathbf{O}(n)$ which is isomorphic to $\otimes_{k=1}^n\mathbf{Z}_2$. 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 $\Gamma(A) \simeq \prod_{i = 1}^k \mathbf{O}(m_i)$ where $m_1,\ldots,m_k$ are the multiplicities of the eigenvalues $\lambda_1,\ldots,\lambda_k$ 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.