  This paper introduces two explicit schemes to sample matrices from Gibbs
distributions on $\mathcal S^{n,p}_+$, the manifold of real positive
semi-definite (PSD) matrices of size $n\times n$ and rank $p$. Given an energy
function $\mathcal E:\mathcal S^{n,p}_+\to \mathbb{R}$ and certain Riemannian
metrics $g$ on $\mathcal S^{n,p}_+$, these schemes rely on an Euler-Maruyama
discretization of the Riemannian Langevin equation (RLE) with Brownian motion
on the manifold. We present numerical schemes for RLE under two fundamental
metrics on $\mathcal S^{n,p}_+$: (a) the metric obtained from the embedding of
$\mathcal S^{n,p}_+ \subset \mathbb{R}^{n\times n} $; and (b) the
Bures-Wasserstein metric corresponding to quotient geometry. We also provide
examples of energy functions with explicit Gibbs distributions that allow
numerical validation of these schemes.
