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 $$\operatorname{det}\left(\mathbb{I}_n + \mathbf{u v}^T\right) = 1 + \langle \mathbf{v}, \mathbf{u} \rangle.$$

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.

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 $$\operatorname{det}\left(A + \mathbf{u v}^T\right) = \operatorname{det}(A)\left(1 + \left\langle \mathbf{v}, A^{-1} \mathbf{u} \right\rangle\right).$$

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.

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 $\left\langle \mathbf{v}, A^{-1} \mathbf{u} \right\rangle \neq -1$.

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 $$J(p) = \sum_{j=1}^{n} f_j(S)^\top \left(C_j^\top C_j - B_j^\top B_j\right) f_j(S)$$

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 $$\left(A + \mathbf{u v}^T\right)^{-1} = A^{-1} - \frac{A^{-1}\mathbf{u}\mathbf{v}^T A^{-1}}{1 + \left\langle \mathbf{v}, A^{-1}\mathbf{u} \right\rangle}$$

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.

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.

Let $B$ and $C$ be two matrices of $\mathbf{GL}_n$. Let $A \in \mathbf{M}_{2n}$ be the block matrix defined as follows: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$
Let $S_1$ be the block matrix $$S_1 = \left(\begin{array}{cc} 0_n & P \\ Q & 0_n \end{array}\right)$$ where $P, Q$ are two elements of $\mathbf{GL}_n$.
Determine the conditions relating $B, C, P, Q$ for the matrices $S_1$ and $S_2 = S_1 A$ to be symmetry matrices.

For every integer $j \in \llbracket 1, n \rrbracket$, we define the matrices $$B_j = S - \alpha_j I_n \quad \text{and} \quad C_j = I_n - \alpha_j S$$
Let $j \in \llbracket 1, n \rrbracket$. Show that $C_j^\top C_j - B_j^\top B_j = (1 - \alpha_j^2) U U^\top$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Let $C \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $\operatorname{det}(C) = 0$. Is it always true that $\operatorname{det}\left(C + \mathbf{u v}^T\right) = 0$? Justify your answer.

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 that $J_n^{(1)}$ admits the following eigenvalues, enumerated with their multiplicity: $$\lambda_k = 2\cos\left(\frac{2\pi k}{n}\right), \quad k \in \llbracket 0, n-1 \rrbracket.$$

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Let $C \in \mathcal{M}_n(\mathbb{R})$ be a matrix such that $\operatorname{det}(C) = 0$. Is it always true that $\operatorname{det}\left(C + \mathbf{u v}^T\right) = 0$? Justify your answer.

12. Deduce that $v _ { n }$ tends to 0.

Let $B$ and $C$ be two matrices of $\mathbf{GL}_n$. Let $A \in \mathbf{M}_{2n}$ be the block matrix defined as follows: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$
Deduce that if $C$ is similar to $B^{-1}$, then $A$ is a product of two symmetry matrices.

Let $B$ and $C$ be two matrices of $\mathbf{GL}_n$. Let $A \in \mathbf{M}_{2n}$ be the matrix defined by blocks as follows: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$ Deduce that if $C$ is similar to $B^{-1}$, then $A$ is a product of two symmetry matrices.

We denote by $D$ the diagonal matrix of size $n$: $$D = \operatorname{Diag}\left((1 - \alpha_j^2)_{1 \leq j \leq n}\right)$$ and $V \in \mathcal{M}_n(\mathbf{R})$ the matrix such that for every $j \in \llbracket 1, n \rrbracket$, the $j$-th column of $V$ is $V_j = f_j(S^\top) U$. Show that $$J(p) = V D V^\top.$$