Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Show that there exists an open set $U$ of $\mathbb{C}[A]$ containing $0$ and an open set $V$ of $\mathbb{C}[A]$ containing the identity matrix $I_n$ such that the exponential function induces a continuous bijection from $U \subset \mathbb{C}[A]$ to $V$ whose inverse is a continuous function on $V$.
Let $A \in \mathscr{M}_n(\mathbb{C})$. We denote by $\mathbb{C}[A]$ the set of elements of $\mathscr{M}_n(\mathbb{C})$ of the form $P(A)$ where $P \in \mathbb{C}[X]$ is a polynomial. We denote
$$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$
Show that there exists an open set $U$ of $\mathbb{C}[A]$ containing $0$ and an open set $V$ of $\mathbb{C}[A]$ containing the identity matrix $I_n$ such that the exponential function induces a continuous bijection from $U \subset \mathbb{C}[A]$ to $V$ whose inverse is a continuous function on $V$.