40 Exponentials of complex matrices

Let $A$ be a $(n\times n)$ complex matrix with eigenvalues ${\lambda }_j$, where for some real $\beta$.

(i) Then the entries of $\exp (tA)$ are complex linear combinations of $t^k{e}^{t{\lambda }_j}$ for integers .

(ii) There exists $M$ such that

The condition means that the point ${\lambda }_j$ lies strictly to the left of the vertical line in the complex plane through $\beta$ on the real axis.