118 Bounded exponentials of matrices

Suppose that $A$ has (not necessarily distinct) eigenvalues such that for all $j=1,\mathrm{\dots },n$. Then there exists $M,\delta >0$ such that

This follows from Corollary 40. The difference between $\mathrm{\Re }\lambda \le 0$ in (iii) of the Proposition 111 and in the Lemma 117 is subtle, and historically important in the theory. Maxwell realised that the stronger hypothesis of the Lemma, strict inequality, is needed to cover the case of multiple eigenvalues, and deal with resonance.