  Let $v_1$,..., $v_n$ be $n$ vectors in an inner product space. Can we find a
natural number $d$ and positive (semidefinite) complex matrices $A_1$,...,
$A_n$ of size $d \times d$ such that ${\rm Tr}(A_kA_l)= <v_k, v_l>$ for all
$k,l=1,..., n$? For such matrices to exist, one must have $<v_k, v_l> \geq 0$
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 $(<v_k, v_l>)$ 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.
