Self-adjoint Matrices are Equivariant

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 $\mathrm{\Gamma }\subset \mathbf{O}(n)$ 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.