Rational and real positive semidefinite rank can be different

Abstract

Given a p×q nonnegative matrix M, the psd rank of MM is the smallest integer k such that there exist k×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_j 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.