prove the existance of $U$ and $V$ Neighbourhoods of $X_0$ and $F(X_0)$ so that $F_{|U}$ is a diffeomorphism in $V$

Question

we have $E = M_n(R)$ and :

$$F : E \to E , F(X) = X^2 + X - I$$

we need to prove the existance of $U\in \mathscr V(X_0)$ and $V\in > \mathscr V(0_E)$ so that $F_{|U}$ (restriction function) is a diffeomorphism from $U$ to $V$ .

with $X_0 = \alpha I / \alpha = \frac{( \sqrt5 - 1 )}{2} $ so we have $F(X_0)=0_E$

i also proved in another question that $F\in C^1$ in $E$ but i dont know how to start anything here !

Answer 1

It suffices to apply the local inversion theorem.