In this part, $a$ denotes an endomorphism of $\mathbf { C } ^ { n }$.
Show that there exist a non-zero natural number $r$, distinct complex numbers $\lambda _ { 1 } , \lambda _ { 2 } , \ldots$, $\lambda _ { r }$, and non-zero natural numbers $m _ { 1 } , m _ { 2 } , \ldots , m _ { r }$, such that: $$\mathbf { C } ^ { n } = \bigoplus _ { i = 1 } ^ { r } E _ { i }$$ where for $i \in \llbracket 1 ; r \rrbracket , E _ { i } = \operatorname { Ker } \left( a - \lambda _ { i } id _ { \mathbf { C } ^ { n } } \right) ^ { m _ { i } }$.

Let $u = (u_k)_{k \geqslant 0}$ be a sequence of $\mathbb{C}$ such that $\mathbb{M}_n(u) \neq \emptyset$. Let $A \in \mathbb{M}_n(u)$. Let $P \in \mathscr{V}(A)$. Show that $\varphi_A$ divides $P$.

For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Deduce an annihilating polynomial of $C_{n,1}$, then its spectrum.