On Subgradients of Spectral Functions

Let $F:\mathbf{S}(m)\rightarrow\overline{\mathbb{R}}$ be a spectral function (i.e. $\mathbf{S}(m)$ is the space of $m\times m$ real symmetric matrices, $\forall O\in\mathbf{O}(m),\forall X\in\mathbf{S}(m),\ F(OX{^tO})=F(X)$, where $\mathbf{O}(m)$ is the orthogonal group and ${^tO}$ is the transpose of $O$). We associate to it the symmetric function $s_F:\mathbb{R}^m\rightarrow\overline{\mathbb{R}}$ by restricting it to the subspace of diagonal matrices. In this work, on the one hand, we give a new, natural proof of the formula which binds the Fréchet subgradients of a spectral function $F$ and the Fréchet subgradients of the function $s_F$ (identical formulas follow for the subgradients and the horizon subgradients); on the other hand we deduce from the previous results and from convexity arguments that, in the general case, a similar formula holds for the Clarke subgradients.