Give a bound for $H_n$ in the case where $J_n$ is moreover an orthogonal matrix distinct from $\pm I_n$.

Let $\mathbf{u}, \mathbf{v}, \mathbf{x}, \mathbf{y} \in \mathbb{R}^n \backslash \{\mathbf{0}\}$. Show that $\mathbf{u v}^T = \mathbf{x y}^T$ if and only if there exists $\lambda \in \mathbb{R} \backslash \{0\}$ such that $$\mathbf{u} = \lambda \mathbf{x}, \quad \text{and} \quad \mathbf{v} = \frac{1}{\lambda} \mathbf{y}.$$

Cyclic subspaces Let $r$ be a nonzero natural integer. Prove that the smallest vector subspace $\mathcal{D}_r$ of $\mathcal{D}$ containing $X^{-r}$ and stable by $\xi$ admits as basis $(X^{k-r})_{0 \leqslant k \leqslant r-1}$. Write the matrix of the endomorphism $\xi_{\mathcal{D}_r}$ induced by $\xi$ on $\mathcal{D}_r$ in this basis.

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.
(b) Show that $K^2 = \operatorname{Tr}(K) K$.
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.

We denote by $J_n^{(\mathrm{C})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ whose coefficients all equal $\frac{1}{n}$, except for its diagonal coefficients, which are zero. Each particle thus interacts in the same way with all other particles.
Determine the spectrum of $U_n = nJ_n^{(\mathrm{C})} + I_n$, then that of $J_n^{(\mathrm{C})}$.

Let $K \in \mathcal{M}_n(\mathbb{R})$ be a matrix of rank 1, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$ be such that $K = \mathbf{u v}^T$.
(a) Show that $\operatorname{Tr}(K) = \langle \mathbf{v}, \mathbf{u} \rangle$.
(b) Show that $K^2 = \operatorname{Tr}(K) K$.
(c) Deduce that $K$ is diagonalizable if and only if $\operatorname{Tr}(K) \neq 0$.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Let $x$ be a nonzero real number. Express $\det(x I_n - A)$ in terms of $x$, $\det A$ and $\det\left(\frac{1}{x} I_n - A^{-1}\right)$.

Compatible extension with $u$ given by a vector Let $V$ be a finite-dimensional vector space equipped with a nilpotent endomorphism $u$. We assume that there exists a vector subspace $W$ of $V$ stable by $u$ and a linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. In this question, we assume that $W$ is strictly contained in $V$ and we fix a vector $v$ of $V$ that does not belong to $W$.
a) Verify that the set $$\mathcal{J} = \{P \in \mathbb{C}[X],\, P(u)(v) \in W\}$$ is an ideal of $\mathbb{C}[X]$.
b) Prove that there exists a natural integer $n$ such that $X^n \in \mathcal{J}$. Deduce that $\mathcal{J}$ is generated by the monomial $X^r$ for an appropriate natural integer $r$ that we do not ask you to specify.
c) Let $W'$ be the subspace of $V$ defined by $$W' = \{P(u)(v) + w,\, P \in \mathbb{C}[X] \text{ and } w \in W\}.$$ Verify that $W'$ contains $W$ and $v$ and that it is stable by $u$.
We denote $G_v = \varphi(u^r(v))$.
d) Prove that there exists an element $F_v$ of $\mathcal{D}$ such that $$G_v = \xi^r(F_v).$$
e) Let $P$ be a polynomial and let $w$ be an element of $W$. Prove that if $P(u)(v) = w$, then $P(\xi)(F_v) = \varphi(w)$.
f) Let $x$ be an element of $W'$. Let $P$ be a polynomial and let $w$ be an element of $W$ such that $x = P(u)(v) + w$. Prove that the element $\varphi'(x) = P(\xi)(F_v) + \varphi(w)$ depends only on $x$ and not on the choice of $P$ and $w$. Verify then that the application $\varphi'$ thus defined is an extension of $\varphi$ to $W'$ compatible with $u$ (it is not asked to verify that $\varphi'$ is linear, which we will admit).

Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.

Let $P \in \mathcal{M}_n(\mathbb{R})$. Show that $P$ is an orthogonal projector of rank 1 if and only if there exists $\mathbf{y} \in \mathbb{R}^n$ with $\|\mathbf{y}\| = 1$ such that $P = \mathbf{y y}^T$.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. Suppose in this question that $A$ is similar to its inverse. Specify the values that the determinant of $A$ can take, and deduce that $\chi_A$ is either reciprocal or antireciprocal.

Let $A$ be a matrix belonging to $\mathbf{GL}_n$. We assume in this question that $A$ is similar to its inverse. Specify the values that the determinant of $A$ can take, and deduce that $\chi_A$ is either reciprocal or antireciprocal.

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$, we define the rational function $g_j \in E$ by $$g_j = \frac{f_j}{\prod_{i=1}^{n}(1 - \alpha_i X)}$$ and the map $P_j$, which associates to a rational function $f \in E$ the rational function $$P_j(f) = \frac{(1 - \alpha_j X)f - (1 - \alpha_j^2)f(\alpha_j)}{X - \alpha_j}$$
Show that for every $j \in \llbracket 1, n \rrbracket$, the map $P_j$ is an endomorphism of $E$ and determine its kernel.

Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1.
For $n \geqslant 0$ we define the vector $U _ { n } \in \mathbb { C } ^ { d }$ by $U _ { n } = \left( u _ { n } , \ldots , u _ { n + d - 1 } \right)$ (recall that $U _ { n }$ is identified with a column vector). Show that the sequence $(U _ { n })$ satisfies a recurrence relation of the form $U _ { n + 1 } = A U _ { n } + B$, with $A \in \mathrm { M } _ { d } ( \mathbb { C } )$ and $B \in \mathbb { C } ^ { d }$ are elements that we shall specify.

Extension to $V$ compatible with $u$ Let $V$ be a finite-dimensional vector space equipped with a nilpotent endomorphism $u$. We assume that there exists a vector subspace $W$ of $V$ stable by $u$ and a linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. Prove that $\varphi$ admits an extension $\psi$ to $V$ compatible with $u$.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Calculate the block matrix product $$\left(\begin{array}{cc} \mathbb{I}_n & 0 \\ \mathbf{v}^T & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n + \mathbf{u}\mathbf{v}^T & \mathbf{u} \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n & 0 \\ -\mathbf{v}^T & 1 \end{array}\right)$$

We denote by $J_n^{(\mathrm{s})}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad J_n^{(\mathrm{S})}(i,j) = \frac{2}{\sqrt{2n+1}} \sin\left(\frac{2\pi ij}{2n+1}\right).$$
Deduce that $J_n^{(\mathrm{s})}$ is a symmetric orthogonal matrix.

Let $A \in \mathrm{GL}_n(\mathbb{R})$ be an invertible matrix, and let $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$. Calculate the block matrix product $$\left(\begin{array}{cc} \mathbb{I}_n & 0 \\ \mathbf{v}^T & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n + \mathbf{u v}^T & \mathbf{u} \\ 0 & 1 \end{array}\right) \left(\begin{array}{cc} \mathbb{I}_n & 0 \\ -\mathbf{v}^T & 1 \end{array}\right).$$

Let $B \in \mathbf{M}_n$ be a diagonalizable matrix. Suppose that the characteristic polynomial of $B$ is reciprocal or antireciprocal. Prove that $B$ is invertible and similar to its inverse.

Let $B \in \mathbf{M}_n$ be a diagonalizable matrix. We assume that the characteristic polynomial of $B$ is reciprocal or antireciprocal. Prove that $B$ is invertible and similar to its inverse.

Until the end of part B, we assume that no root of $p$ is stable.
For every $j \in \llbracket 1, n \rrbracket$ and every $g \in E$, compute $P_j\left(\frac{(X - \alpha_j)g}{1 - \alpha_j X}\right)$.

Problem 2, Part 2: Linear recurrence sequences with constant coefficients
We consider a sequence $\left( u _ { n } \right) _ { n \geqslant 0 }$ of complex numbers defined by the data of $u _ { 0 } , \ldots , u _ { d }$ and the linear recurrence relation $$u _ { n + d } = \sum _ { i = 0 } ^ { d - 1 } a _ { i } u _ { n + i } + b ,$$ where the $a _ { i }$ and $b$ are complex numbers. We define $P \in \mathbb { C } [ X ]$ by $P ( X ) = X ^ { d } - \sum _ { i = 0 } ^ { d - 1 } a _ { i } X ^ { i }$ and we assume that all complex roots of $P$ have modulus strictly less than 1. The matrix $A \in \mathrm{M}_d(\mathbb{C})$ is as defined in question 7.
Calculate the characteristic polynomial of the matrix $A$ (one may reason by induction on $d$).

Splitting of a maximal cyclic subspace Let $V$ be a finite-dimensional vector space over $\mathbb{C}$ and let $u$ be an endomorphism of $V$. We assume that $u$ is nilpotent of index $n$, that is $u^n = 0$ and $u^{n-1} \neq 0$. We choose a vector $v_0$ such that $u^{n-1}(v_0)$ is nonzero.
a) 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.
b) Prove that there exists an injective linear application $\varphi : W \rightarrow \mathcal{D}$ such that $\xi \circ \varphi = \varphi \circ u_W$. According to Part III, this linear application $\varphi$ admits an extension $\psi : V \rightarrow \mathcal{D}$ compatible with $u$.
c) Prove that the kernel of $\psi$ is a complement of $W$ stable by $u$.

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