On the equivariance properties of self-adjoint matrices

We investigate self-adjoint matrices 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 which is isomorphic to . 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 where are the multiplicities of the eigenvalues 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.