  In this short note we prove that a matrix $A\in\mathbb{R}^{n,n}$ is
self-adjoint if and only if it is equivariant with respect to the action of a
group $\Gamma\subset {\bf O}(n)$ which is isomorphic to
$\otimes_{k=1}^n\mathbf{Z}_2$. Moreover we discuss potential applications of
this result, and we use it in particular for the approximation of higher order
derivatives for smooth real valued functions of several variables.
