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 $n \in \mathbb{N}$.