  Gerstenhaber's theorem states that the dimension of the unital algebra
generated by two commuting $n\times n$ matrices is at most $n$. We study the
analog of this question for positive matrices with a positive commutator. We
show that the dimension of the unital algebra generated by the matrices is at
most $\frac{n(n+1)}{2}$ and that this bound can be attained. We also consider
the corresponding question if one of the matrices is a permutation or a
companion matrix or both of them are idempotents. In these cases, the upper
bound for the dimension can be reduced significantly. In particular, the unital
algebra generated by two semi-commuting positive idempotent matrices is at most
$9$-dimensional. This upper bound can be attained.
