# Isotropic functions revisited

## TL;DR

This paper revisits isotropic functions, exploring their derivatives and regularity properties, and extends known relations between scalar functions and their associated operator functions in the context of symmetric matrices.

## Contribution

It extends relations between derivatives of scalar functions and their associated operator functions to a broader setting, highlighting differences in regularity properties.

## Key findings

- Relations between derivatives of $f$ and $F$ are established.
- Regularity properties of $F_{g}$ do not necessarily extend to $F$.
- Provides an example illustrating the difference in regularity properties.

## Abstract

To a smooth and symmetric function $f$ defined on a symmetric open set $\Gamma\subset\mathbb{R}^{n}$ and a real $n$-dimensional vector space $V$ we assign an associated operator function $F$ defined on an open subset $\Omega\subset\mathcal{L}(V)$ of linear transformations of $V$, such that for each inner product $g$ on $V$, on the subspace $\Sigma_{g}(V)\subset\mathcal{L}(V)$ of $g$-selfadjoint operators, $F_{g}=F_{|\Sigma_{g}(V)}$ is the isotropic function associated to $f$, which means that $F_{g}(A)=f(\mathrm{EV}(A))$, where $\mathrm{EV}(A)$ denotes the ordered $n$-tuple of real eigenvalues of $A$. We extend some well known relations between the derivatives of $f$ and each $F_{g}$ to relations between $f$ and $F$. By means of an example we show that well known regularity properties of $F_{g}$ do not carry over to $F$.