We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. We denote by $S p ( A )$ the complex spectrum of $A$. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of: $$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$ Show that: $$\forall x _ { 0 } \in \mathbf { R } ^ { n } , \quad \lim _ { t \rightarrow + \infty } \left\| g _ { x _ { 0 } } ( t ) \right\| = 0 \Longleftrightarrow S p ( A ) \subset \mathbf { R } _ { - } ^ { * } + i \mathbf { R } .$$
We consider $u$ an endomorphism of $\mathbf { R } ^ { n }$, and $A \in \mathscr { M } _ { n } ( \mathbf { R } )$ its matrix in the canonical basis. We denote by $S p ( A )$ the complex spectrum of $A$. Let $g _ { x _ { 0 } }$ be the unique $\mathcal { C } ^ { 1 }$ solution on $\mathbf { R } _ { + }$ of:
$$\left\{ \begin{aligned} y ^ { \prime } & = u ( y ) \\ y ( 0 ) & = x _ { 0 } \end{aligned} \right.$$
Show that:
$$\forall x _ { 0 } \in \mathbf { R } ^ { n } , \quad \lim _ { t \rightarrow + \infty } \left\| g _ { x _ { 0 } } ( t ) \right\| = 0 \Longleftrightarrow S p ( A ) \subset \mathbf { R } _ { - } ^ { * } + i \mathbf { R } .$$