Subgradient of a matrix function related to maximum eigenvalue

Question

Consider the function $f(\textbf{X})$ on dom$f=\mathbb{S}^n$.Show that $$vv^T\in \partial f(\textbf{X})$$ where $v$ is a normalized eigenvector of $\textbf{X}$ associated with $\lambda_{max}(\textbf{X})$, the maximum eigenvalue of $\textbf{X}$. [Hint: Let $f:\mathbb{R}^{m\times n}\to\mathbb{R}$. For $\textbf{X}\in \text{dom} \ f$, a matrix $\textbf{G}_\textbf{X}$ satisfying the following $$f(\textbf{Y})\geq f(\textbf{X})+\text{trace}\{\textbf{G}_\textbf{X}(\textbf{Y}-\textbf{X})\}, \ \forall \textbf{Y}\in \text{dom} \ f$$ is called a subgradient of $f$ at $ \textbf{X}$]

Answer 1

Hint : Observe that \begin{align*} \mathrm{Tr}(vv^T(\mathbf Y-\mathbf X)) &= \mathrm{Tr}(v^T(\mathbf Y-\mathbf X)v) \\&= v^T(\mathbf Y-\mathbf X)v \\&= v^T \mathbf Y v - \lambda_{\max}(\mathbf X)\\&\leq \lambda_{\max}(\mathbf Y) - \lambda_{\max}(\mathbf X) \end{align*}