In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ be the inner product on $\mathbf{R}^n$, and $q$ the associated quadratic form, i.e., $q(x) = b(x,x)$ for all $x \in \mathbf{R}^n$. Prove then that: $$\forall x \in \mathbf { R } ^ { n } , \quad d q ( x ) ( a ( x ) ) = 2 b ( x , a ( x ) ) = - \| x \| ^ { 2 }$$
In this part we consider a map $\varphi$ from $\mathbf { R } ^ { n }$ to $\mathbf { R } ^ { n }$ of class $\mathcal { C } ^ { 1 }$ such that $\varphi ( 0 ) = 0$, and denoting $a = d \varphi ( 0 )$, such that all eigenvalues of $a$ have strictly negative real part. Let $b(x,y) = \int_0^{+\infty} \langle e^{ta}(x) \mid e^{ta}(y) \rangle\, dt$ be the inner product on $\mathbf{R}^n$, and $q$ the associated quadratic form, i.e., $q(x) = b(x,x)$ for all $x \in \mathbf{R}^n$.
Prove then that:
$$\forall x \in \mathbf { R } ^ { n } , \quad d q ( x ) ( a ( x ) ) = 2 b ( x , a ( x ) ) = - \| x \| ^ { 2 }$$