Journal
OPERATIONS RESEARCH LETTERS
Volume 44, Issue 1, Pages 59-60Publisher
ELSEVIER SCIENCE BV
DOI: 10.1016/j.orl.2015.11.012
Keywords
Matrix factorization; Positive semidefinite rank; Semidefinite programming
Categories
Funding
- AFOSR [FA9550-11-1-0305]
- Centre for Mathematics at the University of Coimbra [UID/MAT/00324/2013]
- Fundacao para a Ciencia e a Tecnologia through the European program COMPETE/FEDER
- US NSF Graduate Research Fellowship [DGE-1256082]
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Given a p x q nonnegative matrix M, the psd rank of M is the smallest integer k such that there exist k x k real symmetric positive semidefinite matrices A(1), ... ,A(p) and B-1, ... ,B-q such that M-ij = < A(i) , B-j > for i = 1, ... ,p and j = 1, ... ,q. When the entries of M are rational it is natural to consider the rational restricted psd rank of M, where the factors A(i) and B-i are required to have rational entries. It is clear that the rational-restricted psd rank is always an upper bound to the usual psd rank. We show that this inequality may be strict by exhibiting a matrix with psd rank four whose rational-restricted psd rank is strictly greater than four. (C) 2015 Elsevier B.V. All rights reserved.
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