On commuting matrices and exponentials

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), ×), 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), ×), then the linear subspace Span(S) of M n (C) has property L of Motzkin and Taussky.

Commutativity via spectra of exponentials

“…In fact, the first study of this phenomena goes back to Fréchet [6] and was very quickly followed by [8,9,10,11]. For a more recent development on this matter see [14]. The aim of the current paper is to compare 𝑒 𝑥 𝑒 𝑦 and 𝑒 𝑥+𝑦 with respect to the spectral parameters 𝜌 and #𝜎, but on a larger scale than for single elements 𝑥 and 𝑦.…”

A Method to Construct Sets of Commuting Matrices

“…They appear in a variety of applications in physical and general sciences (McCarthy & Shalit, 2013;Bourgeois, 2013;De Seguins, 2013;Ogata, 2013;Shastry, 2011;Yuzbashyan & Shastry, 2013), where several theoretical and numerical works on differential equations, matrix polynomials equations and general matrix equations get some properties of scalars (Brewer et al, 1986;Gohberg et al, 1982). In such contexts sets of matrices which pairwise commute are needed in numerical experiments.…”