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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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