We consider two distinct complex numbers $\alpha$ and $\beta$. We assume that a matrix $A \in \mathcal{M}_3(\mathbf{C})$ has $\alpha$ as a simple eigenvalue and $\beta$ as a double eigenvalue. $\mathbf{18}$ ▷ Show that $A$ is similar to a matrix of the form $$T = \left( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & a \\ 0 & 0 & \beta \end{array} \right)$$ where $a$ is a certain complex number. Compute $T^n$ for $n$ a natural integer, then $e^{tT}$ for $t$ real. Deduce from this a necessary and sufficient condition on $\alpha$ and $\beta$ for $\lim_{t \rightarrow +\infty} e^{tA} = 0_3$.
We consider two distinct complex numbers $\alpha$ and $\beta$. We assume that a matrix $A \in \mathcal{M}_3(\mathbf{C})$ has $\alpha$ as a simple eigenvalue and $\beta$ as a double eigenvalue.
$\mathbf{18}$ ▷ Show that $A$ is similar to a matrix of the form
$$T = \left( \begin{array}{ccc} \alpha & 0 & 0 \\ 0 & \beta & a \\ 0 & 0 & \beta \end{array} \right)$$
where $a$ is a certain complex number. Compute $T^n$ for $n$ a natural integer, then $e^{tT}$ for $t$ real. Deduce from this a necessary and sufficient condition on $\alpha$ and $\beta$ for $\lim_{t \rightarrow +\infty} e^{tA} = 0_3$.