We are given a power series with complex coefficients $\sum_{n \geqslant 0} a_n z^n$ with radius of convergence $R$ strictly positive, possibly equal to $+\infty$. We denote by $\|\cdot\|$ the norm associated with the inner product $(A,B)\mapsto\operatorname{Tr}({}^tA\times B)$ on $\mathcal{M}_d(\mathbb{R})$.
Let $\mathcal{B} = \left\{A \in \mathcal{M}_d(\mathbb{R}), \|A\| < R\right\}$. Show that the map $\varphi : A \mapsto \varphi(A) = \sum_{n=0}^{+\infty} a_n A^n$ is defined and continuous on $\mathcal{B}$.

We are given a power series with complex coefficients $\sum_{n \geqslant 0} a_n z^n$ with radius of convergence $R$ strictly positive, possibly equal to $+\infty$. We denote by $\|\cdot\|$ the norm associated with the inner product $(A,B)\mapsto\operatorname{Tr}({}^tA\times B)$ on $\mathcal{M}_d(\mathbb{R})$, and $\varphi(A) = \sum_{n=0}^{+\infty} a_n A^n$.
Let $A \in \mathcal{M}_d(\mathbb{R})$ be a nonzero matrix such that $\|A\| < R$.
1) Establish the existence of an integer $r \in \mathbb{N}^*$ such that the family $\left(A^k\right)_{0 \leqslant k \leqslant r-1}$ is free and the family $\left(A^k\right)_{0 \leqslant k \leqslant r}$ is dependent.
2) For $n \in \mathbb{N}$, show the existence and uniqueness of an $r$-tuple $(\lambda_{0,n}, \ldots, \lambda_{r-1,n})$ in $\mathbb{R}^r$ such that $$A^n = \sum_{k=0}^{r-1} \lambda_{k,n} A^k$$
3) Show that there exists a constant $C > 0$ such that: $$\forall n \in \mathbb{N}, \quad \sum_{k=0}^{r-1} |\lambda_{k,n}| \leqslant C \left\|A^n\right\|$$
4) Deduce that, for every integer $k$ between 0 and $(r-1)$, the series $\sum_{n \geqslant 0} a_n \lambda_{k,n}$ is absolutely convergent in $\mathbb{C}$.
5) Conclude that there exists a unique polynomial $P \in \mathbb{R}[X]$ such that $\varphi(A) = P(A)$ and $\deg P < r$.
6) Determine this polynomial $P$ when $A = \begin{pmatrix} 0 & -1 & -1 \\ -1 & 0 & -1 \\ 1 & 1 & 2 \end{pmatrix}$ and $a_n = \frac{1}{n!}$ for all $n \in \mathbb{N}$.

Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$.
Let $M \in \mathcal{B}_{n}$ and $z \in \mathbf{C} \backslash \mathbb{D}$. Show that the series of matrices $\sum \frac{M^{j}}{z^{j+1}}$ converges. We will admit the following fact: let $(E, N)$ be a finite-dimensional normed vector space; if $(v_{j})_{j \in \mathbf{N}}$ is a sequence of elements of $E$ such that the series $\sum N(v_{j})$ converges, then the series $\sum v_{j}$ converges in $E$. If $m \in \mathbf{N}$, give a simplified expression for $\left(zI_{n} - M\right)\sum_{j=0}^{m} \frac{M^{j}}{z^{j+1}}$. Deduce that $$R_{z}(M) = \sum_{j=0}^{+\infty} \frac{M^{j}}{z^{j+1}}$$

Let $A \in \mathscr{M}_n(\mathbb{C})$. Show the equivalence of the following two assertions
(i) $A \in \mathbb{M}_n(v)$ for every sequence $v = (v_k)_{k \geqslant 0}$ of $\mathbb{C}$ satisfying $R_v > 0$.
(ii) $A$ is nilpotent (that is, there exists $k \in \mathbb{N}^*$ such that $A^k = 0_n$).

Show that for every integer $m \geqslant 0$, we have $$D_{u^{(m)}} = D_u$$

Let $v = (v_k)_{k \geqslant 0}$ be another sequence of complex numbers. Show that $$\mathbb{M}_n(u) \cap \mathbb{M}_n(v) \subset \mathbb{M}_n(u+v) \cap \mathbb{M}_n(u \star v)$$

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Show that there exists a continuous map $f$ from $(\mathbb{C}[A])^*$ to $\{0,1\}$ such that $f(M_1) = 0$ and $f(M_2) = 1$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Using the result of question 13a and the path-connectedness of $(\mathbb{C}[A])^*$, conclude that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}$$ Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Show that there exists a continuous map $f$ from $(\mathbb{C}[A])^*$ to $\{0,1\}$ such that $f(M_1) = 0$ and $f(M_2) = 1$.

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 $$(\mathbb{C}[A])^* = \left\{B \in \mathbb{C}[A] \cap \mathrm{GL}_n(\mathbb{C}) \mid B^{-1} \in \mathbb{C}[A]\right\}.$$ We want to show that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$. We suppose that $\exp(\mathbb{C}[A]) \neq (\mathbb{C}[A])^*$ and we fix $M_1, M_2 \in (\mathbb{C}[A])^*$ such that $M_1 \in \exp(\mathbb{C}[A])$ and $M_2 \notin \exp(\mathbb{C}[A])$.
Using the result of question 13a, conclude that $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$.

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. Using the result $\exp(\mathbb{C}[A]) = (\mathbb{C}[A])^*$, conclude that $\exp(\mathscr{M}_n(\mathbb{C})) = \mathrm{GL}_n(\mathbb{C})$.