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arXiv · 1012.4420

On commuting matrices and exponentials

Clément de Seguins Pazzis
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Abstract

Let A and B be matrices of M_n(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 (M_n(C),+) and M -> exp(M) is a homomorphism from G to (GL_n(C),x), then G consists of commuting matrices. If S is a subsemigroup of (M_n(C),+) and M -> exp(M) is a homomorphism from S to (GL_n(C),x), then the linear subspace Span(S) of M_n(C) has property L of Motzkin and Taussky.

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Clément de Seguins Pazzis. 2010-12-20. On commuting matrices and exponentials. https://doi.org/10.1090/s0002-9939-2012-11396-6

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