113 Conclusion of Proof

where $|e^{t\lambda_{j}}|=e^{t\Re\lambda_{j}}\leq e^{t\kappa}$ for all $t\geq 0$ , hence

\sum_{j=1}^{n}|e^{t\lambda_{j}}z_{j}|^{2}\leq e^{2t\kappa}\sum_{j=1}^{n}|z_{j}% |^{2}

so $\|\exp(tD)z\|\leq e^{t\kappa}\|z\|$ ; hence

\|\exp(tA)\|\leq\|S\|\Bigl{\|}\begin{bmatrix}e^{t\lambda_{1}}&0&\dots\\ 0&\ddots&0\\ 0&\dots&e^{t\lambda_{n}}\end{bmatrix}\Bigr{\|}\|S^{-1}\|

\leq\|S\|\|S^{-1}\|\max_{j=1,\dots,n}|e^{t\lambda_{j}}|\leq\|S\|\|S^{-1}\|e^{t% \kappa}.