42 Exponential of a Jordan block

$$J_k(\lambda )=\lambda I_k+N_k$$

where $N_k$ is strictly upper triangular, and $I_k$ and $N_k$ commute, so

$$\exp (t{J}_k(\lambda ))=\exp (t\lambda I_k)\exp (t{N}_k).$$

Now $N_k^k=0$, so we have a polynomial of degree

$$\exp (t{N}_k)=I+t{N}_k+\mathrm{\dots }+t^{k-1}{N}_k^{k-1}/(k-1)!$$

and $|\exp (t\lambda )|=e^{t\mathrm{\Re }\lambda }$, hence we obtain the bound

$$\parallel \exp (t{J}_k(\lambda ))\parallel \le e^{t\mathrm{\Re }\lambda }\left(1+t\parallel N_k\parallel +\mathrm{\dots }+\frac{t^{k-1}{\parallel N_k\parallel }^{k-1}}{(k-1)!}\right).$$