Rational and real positive semidefinite rank can be different

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.