On vector configurations that can be realized in the cone of positive matrices

Let v(1), ... ,v(n) be n vectors in an inner product space. Can we find a dimension d and positive (semidefinite) matrices A(1), ... , A(n) is an element of M-d(C) such that Tr(A(k)A(l)) = < v(k), v(l)> for all k, l = 1, ... , n? For such matrices to exist, one must have < v(k), v(l)> >= 0 for all k, l = 1, ... , n. We prove that if n < 5 then this trivial necessary condition is also a sufficient one and find an appropriate example showing that from n = 5 this is not so - even if we allowed realizations by positive operators in a von Neumann algebra with a faithful normal tracial state. The fact that the first such example occurs at n = 5 is similar to what one has in the well-investigated problem of positive factorization of positive (semidefinite) matrices. If the matrix (< v(k), v(l)>)((k,l)) has a positive factorization, then matrices A(1), ... , A(n) as above exist. However, as we show by a large class of examples constructed with the help of the Clifford algebra, the converse implication is false. (C) 2014 Elsevier Inc. All rights reserved.