Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Show that $D$ is symmetric, with non-negative coefficients and zero diagonal, and has eigenvalues $\lambda_1, \ldots, \lambda_n$, with $\lambda_1$ having eigenspace of dimension 1.

Let $Z \in \mathscr{M}_{d}(\mathbb{R})$ be an invertible matrix with singular value decomposition $Z = VDU^{T}$ where $U = (u_{1}|\ldots|u_{d})$, $V = (v_{1}|\ldots|v_{d})$ are orthogonal matrices and $D = \operatorname{Diag}(\sqrt{\lambda_{1}}, \ldots, \sqrt{\lambda_{d}})$. We assume that $\operatorname{det}(Z) > 0$.

• [(a)] Show that if $R \in \mathrm{SO}_{d}(\mathbb{R})$ then $V^{T}RU \in \mathrm{SO}_{d}(\mathbb{R})$.
• [(b)] Show that $$\sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle Z, R \rangle = \sup_{R \in \mathrm{SO}_{d}(\mathbb{R})} \langle D, R \rangle$$

Let $H$ be a Hadamard matrix of order $n$ with first row constant equal to 1. Let $\lambda_1, \ldots, \lambda_n$ be real numbers such that $$\lambda_1 > 0 \geq \lambda_2 \geq \ldots \geq \lambda_n$$ and $$\sum_{i=1}^{n} \lambda_i = 0.$$ We denote by $U$ the matrix $\frac{1}{\sqrt{n}} H$ and $\Lambda$ the diagonal matrix whose diagonal coefficients are the $\lambda_i$. We finally denote by $D = U^T \Lambda U$.
Show that $D$ is EDM.

Give the value of $\delta(\boldsymbol{x}, \boldsymbol{y})$ as a function of $V_{n}(\boldsymbol{x})$, $V_{n}(\boldsymbol{y})$ and the singular values of $Z(\boldsymbol{x}, \boldsymbol{y})$ in the case where $\operatorname{det}(Z(\boldsymbol{x}, \boldsymbol{y})) > 0$.

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$.

• [(a)] Show that if $\lambda$ is an eigenvalue of $R$ then $\lambda \in \{+1, -1\}$.
• [(b)] Show that $\operatorname{det}(R + I) = \operatorname{det}(R) \operatorname{det}(I + R^{T})$.
• [(c)] Deduce that if $\operatorname{det}(R) = -1$ then $\operatorname{det}(R + I) = 0$.

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$.

• [(a)] Show that there exists an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $u_{d}^{T} Rx = 0$ for all $x \in E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$.
• [(b)] Deduce that $R(E_{1}) \subset E_{1}$ and then that $R(E_{1}) = E_{1}$.

We consider $R \in \mathrm{O}_{d}(\mathbb{R})$ with $\operatorname{det}(R) = -1$, and an orthonormal basis $(u_{1}, \ldots, u_{d})$ of $\mathbb{R}^{d}$ such that $Ru_{d} = -u_{d}$ and $R(E_{1}) = E_{1}$ where $E_{1} = \operatorname{Vect}(u_{1}, \ldots, u_{d-1})$. We consider a matrix $D = \operatorname{Diag}(\alpha_{1}, \ldots, \alpha_{d}) \in \mathscr{M}_{d}(\mathbb{R})$ diagonal with diagonal entries $\alpha_{i} \geqslant 0$ in decreasing order. We denote $U = (u_{1} | \ldots | u_{d})$.

• [(a)] Verify that $\langle D, R \rangle = \langle S, R^{\prime} \rangle$ where $R^{\prime} = U^{T}RU$ and $S = U^{T}DU$.
• [(b)] Show that if $R_{0} = (R_{ij}^{\prime})_{1 \leqslant i,j \leqslant d-1} \in \mathscr{M}_{d-1}(\mathbb{R})$ then $R_{0} \in \mathrm{O}_{d-1}(\mathbb{R})$.

Restriction of a diagonalizable endomorphism to a stable subspace Let $V$ be a finite-dimensional vector space, let $h$ be an endomorphism of $V$ and let $W$ be a subspace stable by $h$. We denote by $h_W$ the endomorphism of $W$ induced by $h$, that is $h_W : W \rightarrow W$, $v \mapsto h(v)$. Prove that if $h$ is diagonalizable, then $h_W$ is also diagonalizable.

Let $V$ be a finite-dimensional vector space, let $h$ be an endomorphism of $V$ and let $W$ be a subspace stable by $h$. We denote by $h_W$ the endomorphism of $W$ induced by $h$, that is $h_W : W \rightarrow W$, $v \mapsto h(v)$. Prove that if $h$ is diagonalizable, then $h_W$ is also diagonalizable.

A matrix invariant For a square matrix $M$ and a nonzero natural integer $k$, we denote $$\delta_k(M) = -\operatorname{dim}\ker M^{k-1} + 2\operatorname{dim}\ker M^k - \operatorname{dim}\ker M^{k+1}.$$
a) Prove that if two square matrices $M$ and $M'$ are similar, then $\delta_k(M) = \delta_k(M')$ for all $k$.
b) Let $r$ be a nonzero natural integer. Verify that for all nonzero integer $k$, $\delta_k(J_r)$ equals 1 if $k = r$ and 0 otherwise.
c) Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$, $$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$ You may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.

We denote by $\lambda_{\text{max}}$ the largest of the eigenvalues of $J_n$ and $\lambda_{\text{min}}$ the smallest. Show that $$\forall x \in \Lambda_n, \quad n\lambda_{\min} \leqslant \sum_{1 \leqslant i,j \leqslant n} J_n(i,j) x_i x_j \leqslant n\lambda_{\max}$$

For a square matrix $M$ and a nonzero natural integer $k$, we denote $$\delta_k(M) = -\operatorname{dim}\ker M^{k-1} + 2\operatorname{dim}\ker M^k - \operatorname{dim}\ker M^{k+1}.$$ Prove that if two square matrices $M$ and $M'$ are similar, then $\delta_k(M) = \delta_k(M')$ for all $k$.

Let $r$ be a nonzero natural integer. Verify that for all nonzero integer $k$, $\delta_k(J_r)$ equals 1 if $k = r$ and 0 otherwise.

Let $M_1$ and $M_2$ be two square matrices and let $M = \operatorname{diag}(M_1, M_2)$. Prove the relation $\operatorname{dim}\ker M = \operatorname{dim}\ker M_1 + \operatorname{dim}\ker M_2$ and then that for all nonzero integer $k$, $$\delta_k(M) = \delta_k(M_1) + \delta_k(M_2).$$ One may use without proof the fact that all these relations extend to a block diagonal matrix $\operatorname{diag}(M_1, \ldots, M_s)$.

The linear application $\widehat{\xi}$ and the endomorphism $\xi$ We denote by $\widehat{\xi} : \mathbb{C}[X^{\pm 1}] \rightarrow \mathcal{D}$ the linear application that to a Laurent polynomial $F$ associates $$\widehat{\xi}(F) = \Pi(XF) \quad \text{and} \quad \xi = \widehat{\xi}_{\mathcal{D}}$$ that is the endomorphism of $\mathcal{D}$ induced by $\widehat{\xi}$.
a) Let $F$ be an element of $\mathbb{C}[X^{\pm 1}]$. Prove that $\widehat{\xi}(\Pi(F)) = \widehat{\xi}(F)$.
b) Let $P$ be a polynomial and let $F$ be an element of $\mathcal{D}$. Prove that $P(\xi)(F) = \Pi(PF)$.

Image and kernel of powers of $\xi$ Let $n$ be a natural integer. Prove that $\xi^n$ is surjective and give a basis of the kernel of $\xi^n$.

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

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.

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 $J_n^{(\mathrm{C})}$.

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).

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 $u$.

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.

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 $u$ 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 $r_1, \ldots, r_s$.

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 $2^\circ$.