  Let M be a p-by-q matrix with nonnegative entries. The positive semidefinite
rank (psd rank) of M is the smallest integer k for which there exist positive
semidefinite matrices $A_i, B_j$ of size $k \times k$ such that $M_{ij} =
\text{trace}(A_i B_j)$. The psd rank has many appealing geometric
interpretations, including semidefinite representations of polyhedra and
information-theoretic applications. In this paper we develop and survey the
main mathematical properties of psd rank, including its geometry, relationships
with other rank notions, and computational and algorithmic aspects.
