A note on Lyapunov transformation and exponential decay in linear ode systems

this paper we consider a class of matrices A where all eigenvalues have neg ative real parts and are of a common magnitude O(1). Concerning the behavio r of e(tA) we provide a necessary and sufficient condition, via Lyapunov tr ansformation, for an estimate of the form \\e(tA)\\2 less than or equal to K-0 exp(-t/K-1) to be valid uniformly for t > 0 with moderate-sized constan ts K-0 and K-1. All relevant relations are quantitatively specified.