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 $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Verify that, in the case where $n = 9$, $J_n^{(1)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge and equals 0 otherwise. This means that each particle interacts only with its two nearest neighbors.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show that $$\operatorname{det}\left(\mathbb{I}_n + \mathbf{u v}^T\right) = 1 + \langle \mathbf{v}, \mathbf{u} \rangle.$$

We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero. Verify that the family $(v_0, u(v_0), \ldots, u^{n-1}(v_0))$ is free and that the subspace $W$ it spans contains $v_0$ and is stable by $u$. Write the matrix of the induced endomorphism $u_W$ in this basis.

Show that the matrix $B = \left(\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 1 \\ 0 & 0 & 0 & \frac{1}{2} \end{array}\right)$ is not similar to its inverse (although its characteristic polynomial $(X-2)^2\left(X-\frac{1}{2}\right)^2$ is reciprocal).
One may determine the eigenspaces of $B$ and $B^{-1}$ for the eigenvalue 2.

Show that the matrix $B = \left(\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 2 & 0 & 0 \\ 0 & 0 & \frac{1}{2} & 1 \\ 0 & 0 & 0 & \frac{1}{2} \end{array}\right)$ is not similar to its inverse (although its characteristic polynomial $\left(X-2\right)^2\left(X-\frac{1}{2}\right)^2$ is reciprocal). One may determine the eigenspaces of $B$ and $B^{-1}$ for the eigenvalue 2.

Until the end of part B, we assume that no root of $p$ is stable.
Deduce that the family $(f_1, \ldots, f_n)$ is linearly independent.

Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show more generally that $$\operatorname{det}\left(A + \mathbf{u}\mathbf{v}^T\right) = \operatorname{det}(A)\left(1 + \left\langle \mathbf{v}, A^{-1}\mathbf{u} \right\rangle\right).$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show more generally that $$\operatorname{det}\left(A + \mathbf{u v}^T\right) = \operatorname{det}(A)\left(1 + \left\langle \mathbf{v}, A^{-1} \mathbf{u} \right\rangle\right).$$

Let $u$ be a nilpotent endomorphism of a finite-dimensional vector space $V$. Prove that there exists a basis of $V$, a natural integer $s$ and nonzero natural integers $r_1 \geqslant \cdots \geqslant r_s$ in which the matrix of $u$ is block diagonal and whose diagonal blocks are Jordan blocks $J_{r_1}, \ldots, J_{r_s}$ of respective sizes $r_1, \ldots, r_s$.

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$.
Prove that if $S_1$ and $S_2$ are two symmetry matrices, the product matrix $A = S_1 S_2$ is invertible and similar to its inverse.

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$. Prove that if $S_1$ and $S_2$ are two symmetry matrices, the product matrix $A = S_1 S_2$ is invertible and similar to its inverse.

Show that the family $\left((S^\top)^i U\right)_{0 \leq i \leq n-1}$ is a basis of $\mathcal{M}_{n,1}(\mathbf{R})$. The matrices $S$ and $U$ were defined in the preliminary part of the problem.

Decomposition theorem: uniqueness of block sizes Prove that the number $s$ and the sizes of the blocks $r_1, \ldots, r_s$ that appear in question $9^\circ$ depend only on $u$ and not on the choice of basis. You may use question $2^\circ$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show that $A + \mathbf{u v}^T$ is invertible if and only if $\left\langle \mathbf{v}, A^{-1}\mathbf{u} \right\rangle \neq -1$.

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 $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Show that, for all $k \in \llbracket 1,n \rrbracket$, $C_{n,1}^k = C_{n,k}$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Show that $A + \mathbf{u v}^T$ is invertible if and only if $\left\langle \mathbf{v}, A^{-1} \mathbf{u} \right\rangle \neq -1$.

Prove that the number $s$ and the sizes of the blocks $r_1, \ldots, r_s$ that appear in question $9^\circ$ depend only on $u$ and not on the choice of basis. One may use question $2^\circ$.

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$.
If a matrix $A$ is a product of two symmetry matrices, is the same true for every matrix similar to $A$?

We say that a matrix $S \in \mathbf{M}_n$ is a symmetry matrix if $S^2 = I_n$. If a matrix $A$ is a product of two symmetry matrices, is the same true for every matrix similar to $A$?

For every integer $j \in \llbracket 1, n \rrbracket$, we define the matrices $$B_j = S - \alpha_j I_n \quad \text{and} \quad C_j = I_n - \alpha_j S$$
Prove that $$J(p) = \sum_{j=1}^{n} f_j(S)^\top \left(C_j^\top C_j - B_j^\top B_j\right) f_j(S)$$

Properties of $h$ In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and the complex number $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$.
a) Prove that $h$ is diagonalizable.
b) Let $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \operatorname{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$.
c) Calculate, for $k$ relative integer, $h^k \circ u \circ h^{-k}$ and, for $l$ natural integer, $h \circ u^l \circ h^{-1}$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Suppose that $A + \mathbf{u v}^T$ is invertible. Show that $$\left(A + \mathbf{u v}^T\right)^{-1} = A^{-1} - \frac{A^{-1}\mathbf{u}\mathbf{v}^T A^{-1}}{1 + \left\langle \mathbf{v}, A^{-1}\mathbf{u} \right\rangle}$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Suppose that $A + \mathbf{u v}^T$ is invertible. Show that $$\left(A + \mathbf{u v}^T\right)^{-1} = A^{-1} - \frac{A^{-1} \mathbf{u v}^T A^{-1}}{1 + \left\langle \mathbf{v}, A^{-1} \mathbf{u} \right\rangle}.$$

Prove that $h$ is diagonalizable.