We extend the work of Kreiss and Morton to prove: for some constant K(m), where m is the order of the matrix A, $|A^(n)v| \leq C(v)$ for all n $geq$ 0 and |v| = 1 implies that $|{SAS}^{-1}| \leq 1$ for some S with $|S^{-1}| \leq 1$, |Sv| $\leq$ k(m)C(v). We establish the analogue for exponentials $e^{Pt}$, and use it to construct the minimal Hilbert norm dominating $L_2$ in which a given partial differential equation with constant coefficients is well-posed.