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 ${\mathrm{\Gamma }}_2(A)\subset \mathbf{O}(n)$ 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 $\mathrm{\Gamma }(A)\simeq {\prod }_{i=1}^k\mathbf{O}(m_i)$ where $m_1,\dots ,m_k$ are the multiplicities of the eigenvalues ${\lambda }_1,\dots ,{\lambda }_k$ 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.