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, $h \circ u^l \circ h^{-1}$.

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.

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: $$A = \left(\begin{array}{cc} B & 0_n \\ 0_n & C \end{array}\right)$$ 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 $C_j^\top C_j - B_j^\top B_j = (1 - \alpha_j^2) U U^\top$.

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 $h$.

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.

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.

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 $W'$. Verify that $u$ and $p$ commute.

We denote $$\bar{p} = \frac{1}{N} \sum_{k=0}^{N-1} h^k \circ p \circ h^{-k}.$$ Prove that the image of $\bar{p}$ is contained in $W$ and that for $w$ in $W$, we have $\bar{p}(w) = w$.

Deduce that $\bar{p}$ is a projector and that its image is $W$.

Prove carefully that $\bar{p}$ commutes with $u$ and $h$.

Deduce that the kernel of $\bar{p}$ is a complement of $W$ and that it is stable under $u$ and $h$.

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 $$J(p) = V D V^\top.$$

``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 $(J_r, D_{r,a})$.

We now consider the case where $A \in \mathcal{S}_n(\mathbb{R})$ is symmetric. Let $\mathbf{u} \in \mathbb{R}^n$ be such that $\|\mathbf{u}\| = 1$. We set $B = A + \mathbf{u}\mathbf{u}^T$. Show that $B \in \mathcal{S}_n(\mathbb{R})$.

Let $(u,v,r,s) \in (\mathbb{N}^*)^4$. Let $A = (a_{ij})_{\substack{1 \leqslant i \leqslant u \\ 1 \leqslant j \leqslant v}} \in \mathcal{M}_{u,v}(\mathbb{R})$ and $B \in \mathcal{M}_{r,s}(\mathbb{R})$. We define the Kronecker product of $A$ by $B$, and we denote $A \otimes B$, the matrix of $\mathcal{M}_{ur,vs}(\mathbb{R})$ which is defined by $uv$ blocks of size $r \times s$ in such a way that, for all $(i,j) \in \llbracket 1,u \rrbracket \times \llbracket 1,v \rrbracket$, the block with index $(i,j)$ is $a_{i,j}B$.
Besides $u,v,r$ and $s$, we are also given two non-zero natural integers, $w$ and $t$.
Show that $\otimes$ is a bilinear map from $\mathcal{M}_{u,v}(\mathbb{R}) \times \mathcal{M}_{r,s}(\mathbb{R})$ to $\mathcal{M}_{ur,vs}(\mathbb{R})$.

We consider the case where $A \in \mathcal{S}_n(\mathbb{R})$ is symmetric. Let $\mathbf{u} \in \mathbb{R}^n$ be such that $\|\mathbf{u}\| = 1$. We set $B = A + \mathbf{u u}^T$. Show that $B \in \mathcal{S}_n(\mathbb{R})$.

Let $n \geq 1$ be an integer and $P \subset \mathbb{R}^n$ a polytope. Show that $P$ has a finite number of faces and at least one vertex.

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.

Let $E$ be a $\mathbb{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$.
Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.

Let $E$ be a $\mathbf{C}$-vector space of dimension $n$. Let $g$ be an endomorphism of $E$ such that $g^n = 0$ and $g^{n-1} \neq 0$. Prove that there exists a basis of $E$ in which the matrix of $g$ is the matrix $N$ below: $$N = \left(\begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & & \ddots & 1 \\ 0 & \ldots & & \ldots & 0 \end{array}\right)$$ In other words: $N = (n_{i,j})_{1 \leq i,j \leq n}$ with $n_{i,j} = 1$ if $j = i+1$ and $n_{i,j} = 0$ otherwise.