In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$ a square matrix with complex coefficients, and we denote by $u$ the endomorphism of $\mathbf{C}^n$ canonically associated with this matrix. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$. For all real $t$ and for $(i,j) \in \llbracket 1,n \rrbracket^2$, we denote by $v_{i,j}(t)$ the coefficient with indices $(i,j)$ of the matrix $e^{tA}$. $\mathbf{19}$ ▷ Show that, if $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$, then $\alpha < 0$.
In all that follows, $\mathbf{K = C}$. We set $E = \mathbf{C}^n$. We are given $A \in \mathcal{M}_n(\mathbf{C})$ a square matrix with complex coefficients, and we denote by $u$ the endomorphism of $\mathbf{C}^n$ canonically associated with this matrix. We set $\alpha = \max_{\lambda \in \operatorname{Sp}(A)} \operatorname{Re}(\lambda)$. For all real $t$ and for $(i,j) \in \llbracket 1,n \rrbracket^2$, we denote by $v_{i,j}(t)$ the coefficient with indices $(i,j)$ of the matrix $e^{tA}$.
$\mathbf{19}$ ▷ Show that, if $\lim_{t \rightarrow +\infty} f_A(t) = 0_n$, then $\alpha < 0$.