I'm interested in (efficiently) sampling from the following density:
$$ p(X) \propto \exp\Big(\sum_{i=1}^n \lambda_i(X)\Big),\quad\textrm{on}\ \{X \in \textrm{Sym}(n) : \lVert X \rVert_F \le 1\}, $$ where $\textrm{Sym}(n)$ denotes the vector space of $n \times n$ symmetric matrices.
The simplest approach would be to use rejection sampling with a rectangular proposal function equal to the maximum of the right-hand side. But even that maximum I'm not sure how to find, given that the constraint that gives the domain translates to $\sum_{i=1}^n \lambda_i^2(X) \le 1$ while inside the $\exp(\cdot)$ the eigenvalues are not squared.
Any pointers to how to approach this are useful. An intermediate question would be, what can we say about the random variable $\Lambda = \sum_{i=1}^n \lambda_i(X)$ when $X$ is drawn uniformly from that norm-constrained set?