We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$.
Deduce that $A$ is $F$-singular if and only if the matrix $$A_{N} = \left(\begin{array}{ccc} A & N_{1} & N_{2} \\ N_{1}^{\top} & 0 & 0 \\ N_{2}^{\top} & 0 & 0 \end{array}\right) = \left(\begin{array}{cc} A & N \\ N^{\top} & 0_{2} \end{array}\right) \in \mathcal{M}_{n+2}(\mathbb{R})$$ is singular.

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Deduce that $\operatorname{det}\left(A_{N}\right) = \operatorname{det}\left(N^{\top} A^{-1} N\right) \operatorname{det}(A)$.

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that there exists $P \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}\left(P^{\top} A^{-1} P\right) = 0$ if and only if there exists $P^{\prime} \in \mathcal{G}_{n,2}(\mathbb{R})$ such that $\operatorname{det}\left(P^{\prime\top} A P^{\prime}\right) = 0$.

We assume $n \geq 3$. Let $F$ be a vector subspace of $E_{n}$ of dimension $n-2$. We consider $(N_1, N_2)$ a basis of $F^{\perp}$ and we set $N = \left(\begin{array}{ll} N_{1} & N_{2} \end{array}\right) \in \mathcal{M}_{n,2}(\mathbb{R})$. $A$ is an invertible matrix in $\mathcal{M}_{n}(\mathbb{R})$.
Show that if $N^{\prime} = \left(\begin{array}{ll} N_{1}^{\prime} & N_{2}^{\prime} \end{array}\right)$ then $$\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) = \left(N_{1}^{\prime\top} A_{s} N_{1}^{\prime}\right)\left(N_{2}^{\prime\top} A_{s} N_{2}^{\prime}\right) - \left(N_{1}^{\prime\top} A_{s} N_{2}^{\prime}\right)^{2} + \left(N_{1}^{\prime\top} A_{a} N_{2}^{\prime}\right)^{2}$$

Let $F$ be a vector subspace of $E_{n}$ of dimension $n-p$, where $1 \leqslant p \leqslant n-1$. We now assume that $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$. Deduce that $\operatorname{det}\left(N^{\prime\top} A N^{\prime}\right) > 0$.

We denote $Q = A _ { p - 1 } A _ { p - 2 } \cdots A _ { 0 }$. Prove that $\operatorname { det } Q = 1$.

We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $A = \left( a _ { i , j } \right)$ be a matrix in $\mathcal { M } _ { n } ( \mathbb { C } )$ that is upper triangular and invertible. Let $\lambda$ be a non-zero complex number. Assume that, for all $i \in \llbracket 1 , n \rrbracket , \frac { a _ { i , i } } { \lambda } \notin \mathcal { V } _ { p }$. Prove that the matrix $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } A ^ { j }$ is invertible.

We keep all the notations from Parts I and II and assume hypotheses (H1)–(H5). Let $\mathcal{B} = (z_1, \ldots, z_\ell)$, where $\ell = 2m-2$, be a basis of $G$. For any element $u$ of $G$, we denote by $U$ (capital letter) the column vector containing the coordinates of $u$ with respect to the basis $\mathcal{B}$. We denote by $A = [a_{i,j}]_{1 \leq i,j \leq \ell}$ and $B = [b_{i,j}]_{1 \leq i,j \leq \ell}$ the two square matrices whose coefficients are defined by $$\forall 1 \leq i,j \leq \ell, \quad a_{i,j} = (z_i \mid z_j), \quad b_{i,j} = (T(z_i) \mid T(z_j))$$
(a) Let $u, v \in G$. Show that $$(u \mid v) = {}^t U A V, \quad (T(u) \mid T(v)) = {}^t U B V$$ and deduce that $A$ and $B$ are invertible.
(b) Let $\lambda \in \mathbb{R}$. Show that an element $u \in G$ is a solution of $(\mathcal{P}_\lambda)$ if and only if $$(A - \lambda B) U = 0$$ Deduce that $(\mathcal{P}_\lambda)$ admits a non-zero solution if and only if $\operatorname{det}(A - \lambda B) = 0$.
(c) We define the function $\psi$ on $\mathbb{R}$ by $$\forall t \in \mathbb{R}, \psi(t) = \frac{\operatorname{det}(A - tB)}{\operatorname{det}(B)}$$ Show that this function $\psi$ is independent of the choice of basis $\mathcal{B}$.
(d) Justify why we can choose the basis $\mathcal{B}$ so that $B = I_\ell$. Deduce that $\psi$ is a polynomial function and specify its degree.
(e) Show that the polynomial $\psi$ is split over $\mathbb{R}[X]$ and that its roots are either simple or double.
(f) Show that $$\psi(X) = \frac{1}{S(w_1, T^{2m-1}(w_1)) S(w_2, T^{2m-1}(w_2))} Q_1(X) Q_2(X)$$ (justify why necessarily the denominator is non-zero). Deduce that $Q_1$ and $Q_2$ are split over $\mathbb{R}[X]$ and have simple roots.

We place ourselves in the particular case where $E = \mathbb{R}_{2m}[X]$, with $m \geq 2$ a fixed natural integer. This vector space is equipped with the scalar product $$\forall (P,Q) \in E^2, \quad (P \mid Q) = \int_{-1}^{1} P(t)Q(t)\,dt$$ The subspace $G = \mathbb{R}_{2m-1}^0[X]$ (polynomials of degree at most $2m-1$ vanishing at $\pm 1$).
Let $(P_1, \ldots, P_{2m-2})$ be any basis of $G$. We consider the two square matrices $A = [a_{i,j}]_{1 \leq i,j \leq 2m-2}$ and $B = [b_{i,j}]_{1 \leq i,j \leq 2m-2}$ defined by $$a_{i,j} = (P_i \mid P_j), \quad b_{i,j} = (P_i' \mid P_j')$$ Determine the ratio $$\frac{\operatorname{det}(A)}{\operatorname{det}(B)}$$ as a function of $m$.

In this question only, we assume $n = 2$, and we denote $$I = \left(\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right) \quad \text{and} \quad J = \left(\begin{array}{cc} 1 & 1 \\ 1 & -1 \end{array}\right)$$ Calculate $S(I)$ and $S(J)$, and deduce $S(A)$ for all $A \in \mathcal{M}_{2}(\{-1,1\})$.

Let $A \in \mathcal{M}_{n}(\{-1,1\})$. Show that the following statements are equivalent:
(a) $n^{2} \in S(A)$.
(b) There exist $X$ and $Y$ in $\{-1,1\}^{n}$ such that $A = X\,{}^{t}Y$.
(c) $A$ is a rank 1 matrix.

Deduce the proportion, among matrices of $\mathcal{M}_{n}(\{-1,1\})$, of matrices $A$ that satisfy $n^{2} \in S(A)$.

We consider three strictly positive integers $n , p$ and $k$ such that $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ is non-empty. Let $A$ be a matrix of $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.
Show that $k \leqslant \min ( n , p )$ and that for all $\lambda \in \mathbb { R } ^ { * } , \lambda A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.

Let $n , p$ and $k$ be three strictly positive integers such that $k \leqslant \min ( n , p )$. Let $A \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ be a matrix of rank $k$ and $( U , \Sigma , V ) \in \mathscr { E }$ such that $A = U \Sigma V ^ { \mathrm { T } }$, $U ^ { \mathrm { T } } U = V ^ { \mathrm { T } } V = I _ { k }$ and $\Sigma$ diagonal with strictly positive diagonal coefficients.
Let $( \bar { U } , \bar { \Sigma } , \bar { V } ) \in \mathscr { E }$. We consider the curve $\gamma : \mathbb { R } \rightarrow \mathscr { M } _ { n , p } ( \mathbb { R } )$ defined by $\gamma ( t ) = ( U + t \bar { U } ) ( \Sigma + t \bar { \Sigma } ) ( V + t \bar { V } ) ^ { \mathrm { T } }$.
(a) Show that the functions $t \mapsto \operatorname { rg } ( U + t \bar { U } ) , t \mapsto \operatorname { rg } ( \Sigma + t \bar { \Sigma } )$ and $t \mapsto \operatorname { rg } ( V + t \bar { V } )$ are constant in a neighborhood of $t = 0$.
(b) Deduce that $\gamma ( t ) \in \mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$ in a neighborhood of $t = 0$.
(c) Show that $\gamma$ is infinitely differentiable on $\mathbb { R }$ and give the expression of the derivative $\gamma ^ { \prime } ( 0 )$ of $\gamma$ at 0.

We denote $A = \left(\begin{array}{ccc} 1 & 3 & -7 \\ 2 & 6 & -14 \\ 1 & 3 & -7 \end{array}\right)$ and $u$ the endomorphism of $\mathbb{C}^3$ canonically associated with $A$.
Calculate the trace and rank of $A$. Deduce, without any calculation, the characteristic polynomial of $A$. Show that $A$ is nilpotent and give its nilpotency index.

Let $V \in \mathcal{M}_n(\mathbb{C})$ be a nilpotent matrix of index $p$. We propose to study the equation $R^2 = V$.
Show that, if $2p - 1 > n$, then there is no solution.

Let $\alpha$ be a non-zero natural integer. Calculate the rank of $J_\alpha^j$ for every natural integer $j$. Deduce that $J_\alpha$ is nilpotent and specify its nilpotency index.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$ with $\alpha_1 \geqslant \cdots \geqslant \alpha_k$.
Deduce the value of $\alpha_1$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For $j \in \mathbb{N}$, we denote $\Lambda_j = \left\{i \in \llbracket 1, k \rrbracket \mid \alpha_i \geqslant j\right\}$. Prove that $\operatorname{rg}\left(N_\sigma^j\right) = \sum_{i \in \Lambda_j} \left(\alpha_i - j\right)$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Prove that, for every $j \in \mathbb{N}^*$, the integer $d_j = \operatorname{rg}\left(u^{j-1}\right) - \operatorname{rg}\left(u^j\right)$ is equal to the number of blocks $J_{\alpha_i}$ whose size $\alpha_i$ is greater than or equal to $j$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Give the value of the integer $k$, the number of blocks $J_{\alpha_i}$ appearing in $N_\sigma$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For every integer $j$ between 1 and $n$, express the number of blocks $J_{\alpha_i}$ of size exactly equal to $j$.

Let $F$ be a vector subspace of $E$ of dimension $p$ and $\mathcal{A}_{F}$ the set of endomorphisms of $E$ that stabilise $F$, that is $\mathcal{A}_{F} = \{ u \in \mathcal{L}(E) \mid u(F) \subset F \}$.
Show that $\operatorname{dim} \mathcal{A}_{F} = n^{2} - pn + p^{2}$.
One may consider a basis of $E$ in which the matrix of every element of $\mathcal{A}_{F}$ is block triangular.

Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$.
Show that $d \leqslant n^{2} - n + 1$ and conclude.

We fix a $\mathbb{C}$-vector space $E$ of dimension $n \geqslant 2$. Let $\mathcal{A}$ be an irreducible subalgebra of $\mathcal{L}(E)$ (i.e., the only vector subspaces stable by all elements of $\mathcal{A}$ are $\{0\}$ and $E$).
Let $v \in \mathcal{A}$ of rank greater than or equal to 2. Show that there exists $u \in \mathcal{A}$ and $\lambda \in \mathbb{C}$ such that $$0 < \operatorname{rg}(v \circ u \circ v - \lambda v) < \operatorname{rg} v.$$ Consider $x$ and $y$ in $E$ such that the family $(v(x), v(y))$ is free, justify the existence of $u \in \mathcal{A}$ such that $u \circ v(x) = y$ and consider the endomorphism induced by $v \circ u$ on $\operatorname{Im} v$.