On the Equivariance Properties of Self-adjoint Matrices

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.