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 $2^\circ$.

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 $j$ be a natural integer strictly less than $N$. By denoting $V_j = \ker(h - \zeta^j \mathrm{id}_V)$ and $V_N = V_0$, verify that $u(V_j) \subset V_{j+1}$.

Calculate, for $k$ relative integer, $h^k \circ u \circ h^{-k}$ and, for $l$ natural integer, $h \circ u^l \circ h^{-1}$.

Search for a stable complement In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$. Let $W$ be a vector subspace of $V$ stable by $u$ and $h$. We assume that $W$ admits a complement $W'$ stable by $u$ and we seek a complement of $W$ stable by $u$ and $h$. Let $p$ be the projector onto $W$ parallel to $W'$.
a) Verify that $u$ and $p$ commute.
We denote $$\bar{p} = \frac{1}{N}\sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$
b) Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.
c) Deduce that $\bar{p}$ is a projector and that its image is $W$.
d) Prove carefully that $\bar{p}$ commutes with $u$ and $h$.
e) Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.

``Graded'' version of the decomposition theorem In this part, we are given: a finite-dimensional vector space $V$; a nilpotent endomorphism $u$ of $V$; a nonzero natural integer $N$ and $\zeta = \exp\frac{2\mathrm{i}\pi}{N}$; an invertible endomorphism $h$ of $V$ such that $h^N = \operatorname{id}_V$ and $h \circ u \circ h^{-1} = \zeta u$.
a) Let $n$ be the index of $u$, that is, the integer such that $u^{n-1} \neq 0$ and $u^n = 0$. Prove that there exists a vector $v$ such that $v$ is an eigenvector of $h$ and $u^{n-1}(v) \neq 0$.
b) Prove that there exists a basis of $V$ in which the matrices of $u$ and $h$ are block diagonal and the diagonal blocks are respectively of the form $$J_r \quad \text{and} \quad D_{r,a} = \operatorname{diag}(\zeta^a, \zeta^{a+1}, \ldots, \zeta^{a+r-1})$$ for $r \in \mathbb{N}^*$ and $a \in \{0, \ldots, N-1\}$ suitable. We call $(r, a)$ the type of such a pair of matrices $(J_r, D_{r,a})$.

Let $n$ be the index of $u$, that is, the integer such that $u^{n-1} \neq 0$ and $u^n = 0$. Prove that there exists a vector $v$ such that $v$ is an eigenvector of $h$ and $u^{n-1}(v) \neq 0$.

Prove that there exists a basis of $V$ in which the matrices of $u$ and $h$ are block diagonal and the diagonal blocks are respectively of the form $$J_r \quad \text{and} \quad D_{r,a} = \operatorname{diag}(\zeta^a, \zeta^{a+1}, \ldots, \zeta^{a+r-1})$$ for $r \in \mathbb{N}^*$ and $a \in \{0, \ldots, N-1\}$ suitable.

An example In this question, we further assume that $N = 4$ and $\ker(h - \operatorname{id}_V) = \{0\}$. For $j \in \{0, \ldots, 3\}$, we denote $V_j = \ker(h - \zeta^j \operatorname{id}_V)$. According to $11^\circ$b), the data of $u$ is equivalent to the data of the two linear maps $u_1 : V_1 \rightarrow V_2$ and $u_2 : V_2 \rightarrow V_3$ induced by $u$.
a) Verify that $u^3 = 0$.
b) Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the ``graded'' version of the decomposition theorem.
c) Prove that the number of blocks of each type is determined by the data of the three dimensions $d_j = \dim V_j$ ($1 \leqslant j \leqslant 3$) and the three ranks $r_1 = \operatorname{rg} u_1$, $r_2 = \operatorname{rg} u_2$ and $r_{21} = \operatorname{rg}(u_2 \circ u_1)$.

In this question, we further assume that $N = 4$ and $\ker(h - \mathrm{id}_V) = \{0\}$. Verify that $u^3 = 0$.

Construct pairs $(u, h)$ that give rise to six different types of pairs of diagonal blocks $(J_r, D_{r,a})$ in the graded version of the decomposition theorem.

Prove that if a complex number $\lambda$ is an eigenvalue of $M$, then $-\lambda$ is also an eigenvalue of $M$ with the same multiplicity.

Let $\chi_M$ be the characteristic polynomial of $M$. We write it as $\chi_M = X^r Q$ where $r$ is an integer and $Q$ is a polynomial whose constant coefficient is nonzero. Briefly justify that $$\mathbb{C}^{m+n} = \ker M^r \oplus \ker Q(M)$$ and verify that these subspaces are stable under $H$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$.
Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. Show that for all $\mathbf{w} \in \mathbb{R}^n$, we have $\mathbb{E}\left[\langle \mathbf{U}, \mathbf{w} \rangle^2\right] = \frac{1}{n} \|\mathbf{w}\|^2$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$, and for all $x \in \mathbb{R}$, $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$.
Let $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$. Show that the random variable $\chi_B(x)$ has finite expectation, and that, denoting by $\chi_A'$ the derivative of the polynomial $\chi_A$, we have $$\mathbb{E}\left[\chi_B(x)\right] = \chi_A(x) - \frac{1}{n} \chi_A'(x)$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. For all $x \in \mathbb{R}$, we denote $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$. Let $x \in \mathbb{R} \backslash \left\{\lambda_1, \ldots, \lambda_n\right\}$. Show that the random variable $\chi_B(x)$ has finite expectation, and that, denoting by $\chi_A'$ the derivative of the polynomial $\chi_A$, we have $$\mathbb{E}\left[\chi_B(x)\right] = \chi_A(x) - \frac{1}{n} \chi_A'(x).$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$, and for all $x \in \mathbb{R}$, $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$.
Show that for all $k \in \{1, 2, \ldots, n\}$, we have $$\mathbb{E}\left[\chi_B\left(\lambda_k\right)\right] = -\frac{1}{n} \chi_A'\left(\lambda_k\right)$$

In this subsection, we assume that $J_n = J_n^{(1)}$, the matrix introduced in subsection A-IV.
We set $A = \begin{pmatrix} \mathrm{e}^{\beta - h} & \mathrm{e}^{-\beta - h} \\ \mathrm{e}^{-\beta + h} & \mathrm{e}^{\beta + h} \end{pmatrix}$.
Determine the eigenvalues of the matrix $A$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Show that for all $k \in \{1,2,\ldots,n\}$, we have $$\mathbb{E}\left[\chi_B\left(\lambda_k\right)\right] = -\frac{1}{n} \chi_A'\left(\lambda_k\right).$$

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. For $x \in \mathbb{R}$ we denote $\chi_A(x) = \operatorname{det}\left(x \mathbb{I}_n - A\right)$. We consider any orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable defined on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$ taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, and which follows the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$, and for all $x \in \mathbb{R}$, $\chi_B(x) = \operatorname{det}\left(x \mathbb{I}_n - B\right)$.
Prove that there exists $x \in \mathbb{R}$ such that $\mathbb{E}\left[\chi_B(x)\right] \neq 0$.

In this fourth part, $A \in \mathcal{S}_n(\mathbb{R})$ is a symmetric matrix whose eigenvalues are denoted $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$. We consider an arbitrary orthonormal basis $\left(\mathbf{u}_1, \ldots, \mathbf{u}_n\right)$. Let $\mathbf{U}$ be a random variable taking values in the finite set $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_n\right\}$, following the uniform distribution on this set. We consider the random variable $B = A + \mathbf{U}\mathbf{U}^T$. Prove that there exists $x \in \mathbb{R}$ such that $\mathbb{E}\left[\chi_B(x)\right] \neq 0$.

As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$.
Justify that $(A - \varepsilon \mathbb{I}_n)$ is invertible.

As in the third part, we assume that $B = A + \mathbf{u u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further assume that there exists an integer $m \in \{1,2,\ldots,n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. Justify that $(A - \varepsilon \mathbb{I}_n)$ is invertible.

As in the third part, we suppose that $B = A + \mathbf{u}\mathbf{u}^T$ with $A \in \mathcal{S}_n(\mathbb{R})$ a symmetric matrix, and $\mathbf{u} \in \mathbb{R}^n$ a vector such that $\|\mathbf{u}\| = 1$. We denote by $\lambda_1 \leqslant \lambda_2 \leqslant \cdots \leqslant \lambda_n$ the eigenvalues of $A$ and $\mu_1 \leqslant \mu_2 \leqslant \cdots \leqslant \mu_n$ those of $B$. We admit that $$\lambda_1 \leqslant \mu_1 \leqslant \lambda_2 \leqslant \mu_2 \leqslant \cdots \leqslant \lambda_n \leqslant \mu_n.$$ We further suppose that there exists an integer $m \in \{1, 2, \ldots, n-1\}$ such that the eigenvalues of $A$ satisfy $$0 = \lambda_1 = \lambda_2 = \cdots = \lambda_m < \lambda_{m+1} \leqslant \cdots \leqslant \lambda_n.$$ Let $\varepsilon \in ]0, \lambda_{m+1}[$. We suppose that $\left\langle \mathbf{u}, \left(A - \varepsilon \mathbb{I}_n\right)^{-1}\mathbf{u}\right\rangle < -1$.
Show that $(B - \varepsilon \mathbb{I}_n)$ is invertible.