The average density of states in the Gaussian ensembles whose eigenvalues are restricted to lie in a given interval, or to be greater than a given minimal value, was calculated by Dean and Majumdar in <A HREF="https://arxiv.org/abs/0801.1730">Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices</A>. The eigenvalue density has a $1/\sqrt x$ divergence at the end points of the interval. The case in the OP of a constraint of positive eigenvalues is the red solid curve in this plot from the cited paper.

<IMG SRC="https://ilorentz.org/beenakker/MO/Majumdar.png" WIDTH="400"/>