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Rational and real positive semidefinite rank can be different. (English) Zbl 1408.15016
Summary: Given a $$p \times q$$ nonnegative matrix $$M$$, the psd rank of $$M$$ is the smallest integer $$k$$ such that there exist $$k \times k$$ real symmetric positive semidefinite matrices $$A_1, \ldots, A_p$$ and $$B_1, \ldots, B_q$$ such that $$M_{i j} = \langle A_i, B_j \rangle$$ for $$i = 1, \ldots, p$$ and $$j = 1, \ldots, q$$. 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.

##### MSC:
 15B48 Positive matrices and their generalizations; cones of matrices
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