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arXiv · 1603.08413

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

Marko Kandić·Klemen Šivic
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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.

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BibTeXRIS

Marko Kandić, Klemen Šivic. 2016-03-04. On the dimension of the algebra generated by two positive semi-commuting matrices. https://arxiv.org/abs/1603.08413

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