For real, symmetric, positive semidefinite matrices $A$ and $B$, let $\leq_L$ denote the Loewner partial order: $A \leq_L B$ iff $B-A$ is positive semidefinite. Suppose $A$ and $B$ are fixed PSD matrices. Is there a method to randomly generate PSD matrices from the region $\{X | A \leq_L X \leq_L B\}$? Better yet, is there a way to uniformly sample matrices from this region?
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Let $\mathbb S_n$ be the set of $n \times n$ symmetric matrices. Given positive semidefinite matrices $\mathrm A, \mathrm B \in \mathbb S_n$, the following (convex) set
$$\{ \mathrm X \in \mathbb S_n \mid \mathrm A \preceq \mathrm X \preceq \mathrm B \}$$
is a spectrahedron. To sample from spectrahedra, take a look at Narayanan's paper [0] and the references therein.
[0] Hariharan Narayanan, Randomized Interior Point methods for Sampling and Optimization, arXiv:0911.3950.
