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PROPERTY L AND COMMUTING EXPONENTIALS IN DIMENSION AT MOST THREE

Part of: Basic linear algebra

Published online by Cambridge University Press:  28 June 2013

GERALD BOURGEOIS
GERALD BOURGEOIS*
Affiliation:
GAATI, Université de la Polynésie Française, BP 6570, 98702 FAA’A, Tahiti, Polynésie Française email bourgeois.gerald@gmail.com
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Abstract

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Let $A, B$ be two square complex matrices of the same dimension $n\leq 3$. We show that the following conditions are equivalent. (i) There exists a finite subset $U\subset { \mathbb{N} }_{\geq 2} $ such that for every $t\in \mathbb{N} \setminus U$, $\exp (tA+ B)= \exp (tA)\exp (B)= \exp (B)\exp (tA)$. (ii) The pair $(A, B)$ has property L of Motzkin and Taussky and $\exp (A+ B)= \exp (A)\exp (B)= \exp (B)\exp (A)$. We also characterise the pairs of real matrices $(A, B)$ of dimension three, that satisfy the previous conditions.

Keywords

matrix exponentialmatrix pencilproperty L

MSC classification

Secondary: 15A16: Matrix exponential and similar functions of matrices 15A22: Matrix pencils 15A24: Matrix equations and identities
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 1 , February 2014 , pp. 70 - 78
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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