  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.
