Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-2012-11396-6
Published electronically: July 10, 2012
MathSciNet review: 3003670
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • 1. G. Bourgeois, On commuting exponentials in low dimensions,
    Linear Algebra Appl.
    423
    (2007)
    277-286. MR 2312407 (2008d:39030)
  • 2. G. Fischer.
    Plane Algebraic Curves,
    Student Mathematical Library, Volume 15,
    AMS, 2001. MR 1836037 (2002g:14042)
  • 3. M. Fréchet, Les solutions non-commutables de l'équation matricielle $ e^{x+y}=e^xe^y$,
    Rend. Circ. Math. Palermo
    2
    (1952)
    11-27. MR 0049857 (14:237a)
  • 4. M. Fréchet, Les solutions non-commutables de l'équation matricielle $ e^{x+y}=e^xe^y$, Rectification,
    Rend. Circ. Math. Palermo
    2
    (1953)
    71-72. MR 0057836 (15:279h)
  • 5. N.J. Higham.
    Functions of Matrices. Theory and Computation,
    SIAM, 2008. MR 2396439 (2009b:15001)
  • 6. C.W. Huff, On pairs of matrices (of order two) $ A,B$ satisfying the condition $ e^{A+B}=e^Ae^B \neq e^B e^A$,
    Rend. Circ. Math. Palermo
    2
    (1953)
    326-330. MR 0062706 (16:4c)
  • 7. A.G. Kakar, Non-commuting solutions of the matrix equation $ \exp (X+Y)=\exp (X)\exp (Y)$,
    Rend. Circ. Math. Palermo
    2
    (1953)
    331-345. MR 0062708 (16:4e)
  • 8. T. Kato.
    Perturbation Theory for Linear Operators,
    Grundlehren der Mathematischen Wissenschaften, Second edition, Springer-Verlag, 1976. MR 0407617 (53:11389)
  • 9. K. Morinaga, T. Nono, On the non-commutative solutions of the exponential equation $ e^xe^y = e^{x+y}$,
    J. Sci. Hiroshima Univ.
    (A)17
    (1954)
    345-358. MR 0066337 (16:558f)
  • 10. K. Morinaga, T. Nono, On the non-commutative solutions of the exponential equation $ e^xe^y = e^{x+y}$ II,
    J. Sci. Hiroshima Univ.
    (A)18
    (1954)
    137-178. MR 0072104 (17:228h)
  • 11. T.S. Motzkin, O. Taussky, Pairs of matrices with property L,
    Trans. Amer. Math. Soc.
    73
    (1952)
    108-114. MR 0049855 (14:236e)
  • 12. T.S. Motzkin, O. Taussky, Pairs of matrices with property L (II),
    Trans. Amer. Math. Soc.
    80
    (1955)
    387-401. MR 0086781 (19:242c)
  • 13. H. Radjavi, P. Rosenthal,
    Simultaneous Triangularization,
    Universitext, Springer-Verlag (2000). MR 1736065 (2001e:47001)
  • 14. C. Schmoeger, Remarks on commuting exponentials in Banach algebras,
    Proc. Amer. Math. Soc.
    127 (5)
    (1999)
    1337-1338. MR 1476391 (99h:46090)
  • 15. C. Schmoeger, Remarks on commuting exponentials in Banach algebras II,
    Proc. Amer. Math. Soc.
    128 (11)
    (2000)
    3405-3409. MR 1691002 (2001b:46077)
  • 16. E.M.E. Wermuth, Two remarks on matrix exponentials,
    Linear Algebra Appl.
    117
    (1989)
    127-132. MR 993038 (90e:15019)
  • 17. E.M.E. Wermuth, A remark on commuting operator exponentials,
    Proc. Amer. Math. Soc.
    125 (6)
    (1997)
    1685-1688. MR 1353407 (97g:39011)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 15A16, 15A22

Retrieve articles in all journals with MSC (2010): 15A16, 15A22


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: https://doi.org/10.1090/S0002-9939-2012-11396-6
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.

American Mathematical Society