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

TL;DR

This paper investigates the dimension of algebras generated by two positive matrices with a positive commutator, extending Gerstenhaber's theorem, and finds bounds that depend on matrix properties like idempotency or permutation structure.

Contribution

It establishes new upper bounds for the algebra dimension generated by positive semi-commuting matrices, including special cases with idempotent or permutation matrices, and shows these bounds are sharp.

Findings

Dimension at most n(n+1)/2 for positive matrices with positive commutator

Upper bound reduced to 9 for semi-commuting positive idempotent matrices

Bounds are sharp and attainable in specific cases

Abstract

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.