Two remarks on matrix exponentials

Abstract It is shown that for two square matrices A and B with algebraic elements, e A e B = e B e A if and only if AB = BA . Moreover, the in some sense best possible perturbation inequality ‖e A − e B ‖ 2 ⩽ ‖e A ‖ 2 (e ‖A − B‖ 2 − 1) for a normal matrix A is proven.