# Self-adjoint Matrices are Equivariant

## TL;DR

This paper proves that self-adjoint matrices are exactly those matrices that are equivariant under a specific group action, and explores applications in approximating higher order derivatives of smooth functions.

## Contribution

It establishes a novel characterization of self-adjoint matrices via group equivariance and discusses potential applications in differential approximation.

## Key findings

- Self-adjoint matrices are equivariant under a specific group action.
- The result links matrix symmetry to group theory.
- Applications include approximating higher order derivatives.

## Abstract

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.