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

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• 作者：Marko Kandić ^{marko.kandic@fmf.uni-lj.si" class="auth_mail" title="E-mail the corresponding author} ; Klemen &Scaron ; ivic ; ^{klemen.sivic@fmf.uni-lj.si" class="auth_mail" title="E-mail the corresponding author}
• 关键词：15A27 ; 15B48 ; 47B47
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：512
• 期：Complete
• 页码：136-161
• 全文大小：456 K

文摘

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