On the dimension of the algebra generated by two positive semi-commuting matrices

Gerstenhaber's theorem states that the dimension of the unital algebra generated by two commuting n×nn×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 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.