grandes-ecoles 2022 Q24

grandes-ecoles · France · mines-ponts-maths2__mp Invariant lines and eigenvalues and vectors Properties of eigenvalues under matrix operations
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$.
$\mathbf{24}$ ▷ We assume, in this question only, that all eigenvalues of the matrix $A$ have real parts that are positive or zero. Show that, if $X \in \mathbf{C}^n$, we have $$\lim_{t \rightarrow +\infty} e^{tA} X = 0 \Longleftrightarrow X = 0.$$ 
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$.

$\mathbf{24}$ ▷ We assume, in this question only, that all eigenvalues of the matrix $A$ have real parts that are positive or zero. Show that, if $X \in \mathbf{C}^n$, we have
$$\lim_{t \rightarrow +\infty} e^{tA} X = 0 \Longleftrightarrow X = 0.$$
Paper Questions
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