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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On commuting matrices and exponentials


Author: Clément de Seguins Pazzis
Journal: Proc. Amer. Math. Soc. 141 (2013), 763-774
MSC (2010): Primary 15A16; Secondary 15A22
Published electronically: July 10, 2012
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Abstract: Let $ A$ and $ B$ be matrices of $ \operatorname {M}_n(\mathbb{C})$. We show that if
$ \exp (A)^k \exp (B)^l=\exp (kA+lB)$ for all integers $ k$ and $ l$, then $ AB=BA$. We also show that if $ \exp (A)^k \exp (B)=\exp (B)\exp (A)^k= \exp (kA+B)$ for every positive integer $ k$, then the pair $ (A,B)$ has property L of Motzkin and Taussky.

As a consequence, if $ G$ is a subgroup of $ (\operatorname {M}_n(\mathbb{C}),+)$ and $ M\mapsto \exp (M)$ is a homomorphism from $ G$ to $ (\operatorname {GL}_n(\mathbb{C}),\times )$, then $ G$ consists of commuting matrices. If $ S$ is a subsemigroup of $ (\operatorname {M}_n(\mathbb{C}),+)$ and $ M \mapsto \exp (M)$ is a homomorphism from $ S$ to $ (\operatorname {GL}_n(\mathbb{C}),\times )$, then the linear subspace $ \operatorname {Span}(S)$ of $ \operatorname {M}_n(\mathbb{C})$ has property L of Motzkin and Taussky.


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Additional Information

Clément de Seguins Pazzis
Affiliation: Lycée Privé Sainte-Geneviève, 2, rue de l’École des Postes, 78029 Versailles Cedex, France
Email: dsp.prof@gmail.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11396-6
PII: S 0002-9939(2012)11396-6
Keywords: Matrix pencils, commuting exponentials, property L
Received by editor(s): December 30, 2010
Received by editor(s) in revised form: May 24, 2011, and July 16, 2011
Published electronically: July 10, 2012
Communicated by: Gail R. Letzter
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.