111 Growth of exponentials of diagonable matrices

Suppose that $A$ has distinct eigenvalues ${\lambda }_j$ such that $\mathrm{\Re }{\lambda }_j\le \kappa$ for all $j=1,\mathrm{\dots },n$. (i) Then the general solution of $\frac{dX}{dt}=AX$ is

where $X_j$ is an eigenvector corresponding to ${\lambda }_j$ and $a_j\in 𝐂$ are arbitrary constants.

(ii) There exists $M$ such that

(iii) In particular, suppose that $\mathrm{\Re }{\lambda }_j\le 0$ for all $j=1,\mathrm{\dots },n$. Then there exists $M$ such that $\parallel \exp (tA)\parallel \le M\mathit{\hspace{1em}\hspace{1em}}(t\ge 0).$