  We show that a formal power series in $2N$ non-commuting indeterminates is a
positive non-commutative kernel if and only if the kernel on $N$-tuples of
matrices of any size obtained from this series by matrix substitution is
positive. We present two versions of this result related to different classes
of matrix substitutions. In the general case we consider substitutions of
jointly nilpotent $N$-tuples of matrices, and thus the question of convergence
does not arise. In the ``convergent'' case we consider substitutions of
$N$-tuples of matrices from a neighborhood of zero where the series converges.
Moreover, in the first case the result can be improved: the positivity of a
non-commutative kernel is guaranteed by the positivity of its values on the
diagonal, i.e., on pairs of coinciding jointly nilpotent $N$-tuples of
matrices. In particular this yields an analogue of a recent result of Helton on
non-commutative sums-of-squares representations for the class of hereditary
non-commutative polynomials. We show by an example that the improved
formulation does not apply in the ``convergent'' case.
