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Positive semidefinite rank. (English) Zbl 1327.90174
Summary: Let $$M \in \mathbb R^{p \times q}$$ be a nonnegative matrix. 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 $$M_{ij} = \mathrm{trace}(A_i B_j)$$. 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.

##### MSC:
 90C22 Semidefinite programming 15A23 Factorization of matrices 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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##### References:
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