Consider a linear ODE: $\dot{x} = A x$ where $A$ is Hurwitz, i.e. all its eigenvalues have negative real parts. Thus the system is exponentially stable. We know that there exists positive numbers $\beta$ and $\alpha$ such that $\| e^{A t} \| \leq \beta e^{-\alpha t}$ for all $t$. I see this result being used in many analysis.
My question is how to (practically) compute these values? In particular, if I pick $\alpha$ so that $ -\alpha > \max_i \Re(\lambda_i)$, where $\lambda_i$ are the eigenvalues of $A$, then how to compute a tight value for $\beta$?