We fix two symplectic forms $\omega$ and $\omega _ { 1 }$ on $E$. Let $\mathcal { S }$ be the set defined in question 13: $$\mathcal { S } = \left\{ u \in \mathrm { GL } ( E ) : \forall ( x , y ) \in E ^ { 2 } , \omega ( x , u ( y ) ) = \omega ( u ( x ) , y ) \right\}$$ Show that the set of elements of $\mathcal { S }$ whose characteristic polynomial $P$ has roots of multiplicity at most 2 in $\mathbb { C }$ is dense in $\mathcal { S }$.
Hint: You may use $r \left( P ^ { \prime } \right)$ where the map $r$ is defined in question 11.

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 $m$.

Show that for every matrix $A$ in $\mathcal{M}_{n}(\{-1,1\})$, the set $S(A)$ is included in $\{-n^{2}, \ldots, n^{2}\}$. Show that the inclusion is strict (one may think of a parity argument), and show that $S(A)$ is a symmetric set, in the sense that an integer $k$ is in $S(A)$ if and only if $-k$ is in $S(A)$.

Let $A$ and $B$ be in $\mathcal{M}_{n}(\{-1,1\})$. Suppose that there exist diagonal matrices $C$ and $D$ containing only 1s and $-1$s on the diagonal, such that $B = CAD$. Show that $S(A) = S(B)$.

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 $n^{2} \in S(A)$.

For $A = (a_{i,j})_{1 \leqslant i,j \leqslant n} \in \mathcal{M}_{n}(\{-1,1\})$ and $Y = (y_{i})_{1 \leqslant i \leqslant n} \in \{-1,1\}^{n}$, we denote $$g_{A}(Y) = \max\left\{{}^{t}X A Y \mid X \in \{-1,1\}^{n}\right\}.$$
Show that the function $g_{A}$ can be rewritten as $$g_{A}(Y) = \sum_{i=1}^{n} \left|\sum_{j=1}^{n} a_{i,j} y_{j}\right|.$$

We fix $A \in \mathcal{M}_{n}(\{-1,1\})$ and denote $$m(A) := \min(S(A) \cap \mathbb{N}).$$
For $Y \in \{-1,1\}^{n}$, show that we have $$\min\left\{\left|{}^{t}X A Y\right| \mid X \in \{-1,1\}^{n}\right\} \leqslant n$$ and deduce $m(A) \leqslant n$.

We denote by $\mathbb{R}_{N}[X]$ the vector space of polynomials with real coefficients, of degree at most $N$. We define the set $A_{N}$ formed by $P \in \mathbb{R}_{N}[X]$, such that $P(-1) = P(1) = 1$, which furthermore satisfy $P(x) \geqslant 0$ for all $x$ in the interval $[-1,1]$. We define on $\mathbb{R}_{N}[X]$ a linear form $L$ by $L(P) = \int_{-1}^{1} P(x)\,dx$.
(a) Verify that $A_{N}$ is a convex subset of $\mathbb{R}_{N}[X]$.
(b) Show that the expression $$\|P\|_{1} = \int_{-1}^{1} |P(x)|\,dx$$ defines a norm on $\mathbb{R}_{N}[X]$.
(c) Show that $A_{N}$ is closed in the normed vector space $\left(\mathbb{R}_{N}[X], \|\cdot\|_{1}\right)$.

Let $a$ and $b$ be in $E$. Show the following relation and give a geometric interpretation:
$$\|a + b\|^{2} + \|a - b\|^{2} = 2(\|a\|^{2} + \|b\|^{2})$$

Let $a$ and $b$ be in $E$. Show the following relation and give a geometric interpretation:
$$\|a + b\|^{2} + \|a - b\|^{2} = 2(\|a\|^{2} + \|b\|^{2})$$

Show that $\operatorname{Toep}_{n}(\mathbb{C})$ is a vector subspace of $\mathcal{M}_{n}(\mathbb{C})$. Give a basis for it and specify its dimension.

Give the expression of $\langle A , B \rangle _ { F }$ as a function of the coefficients of $A$ and $B$.

Show that if two matrices $A$ and $B$ commute $(AB = BA)$ and if $P$ and $Q$ are two polynomials of $\mathbb{C}[X]$, then $P(A)$ and $Q(B)$ commute.

Let $u \in \mathbb { R } ^ { p }$. Show that $\| A u \| _ { 2 } \leqslant \| A \| _ { F } \| u \| _ { 2 }$.

Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Give the characteristic polynomial of $A$.

Show that $\| A C \| _ { F } \leqslant \| A \| _ { F } \| C \| _ { F }$.

Let $A = \left(\begin{array}{cc} a & b \\ c & a \end{array}\right)$ be a Toeplitz matrix of size $2 \times 2$, where $(a, b, c)$ are complex numbers. Discuss, depending on the values of $(a, b, c)$, the diagonalizability of $A$.

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 $M = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ be a matrix of $\mathcal{M}_{2}(\mathbb{C})$. Show that $M$ is similar to a matrix of type $\left(\begin{array}{cc} \alpha & 0 \\ 0 & \beta \end{array}\right)$ or of type $\left(\begin{array}{cc} \alpha & \gamma \\ 0 & \alpha \end{array}\right)$, where $\alpha, \beta$ and $\gamma$ are complex numbers with $\alpha \neq \beta$.

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 $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$.
Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Verify that $S$ is a symmetric matrix that admits only non-negative eigenvalues and then show that $\operatorname { Im } ( A ) = \operatorname { Im } ( S )$.
(b) Let $u \in \mathbb { R } ^ { n }$ be an eigenvector of $S$ for an eigenvalue $\lambda > 0$ and let $v = A ^ { \mathrm { T } } u / \sqrt { \lambda } \in \mathbb { R } ^ { p }$. Show that $v$ is an eigenvector of $\tilde { S }$ for the eigenvalue $\lambda$ and $\| v \| _ { 2 } = \| u \| _ { 2 }$.

We equip $\mathbb{R}_{n}[X]$ with the inner product defined by $$\langle P, Q \rangle = \int_{-1}^{1} P(x)Q(x)\,dx$$ For $j \in \mathbb{N}$, we define the polynomial $$P_{j}(X) = \frac{1}{2^{j} j!} \frac{d^{j}}{dX^{j}}\left[(X^{2}-1)^{j}\right]$$ By convention, $P_{0} = 1$. The subspace of $\mathbb{R}_{n}[X]$ formed by even polynomials is denoted $\Pi_{n}$, and that of odd polynomials is denoted $J_{n}$.
(a) Show that the family $\left(P_{j}\right)_{0 \leqslant j \leqslant n}$ is a basis of $\mathbb{R}_{n}[X]$.
(b) Deduce that the family $\left(P_{2j}\right)_{0 \leqslant j \leqslant \frac{n}{2}}$ is a basis of $\Pi_{n}$, while the family $\left(P_{2j+1}\right)_{0 \leqslant j \leqslant \frac{n-1}{2}}$ is a basis of $J_{n}$.

Deduce that every matrix of $\mathcal{M}_{2}(\mathbb{C})$ is similar to a Toeplitz matrix.

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 $\mathscr { M } _ { n , p } ^ { k } ( \mathbb { R } )$. Let $S = A A ^ { \mathrm { T } }$ and $\tilde { S } = A ^ { \mathrm { T } } A$.
(a) Show that there exist $U \in \mathscr { M } _ { n , k } ( \mathbb { R } )$ and $\Lambda = \operatorname { diag } \left( \lambda _ { 1 } , \ldots , \lambda _ { k } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$ such that $S = U \Lambda U ^ { \mathrm { T } }$ with $\lambda _ { 1 } \geqslant \ldots \geqslant \lambda _ { k } > 0$ and $U ^ { \mathrm { T } } U = I _ { k }$.
(b) Show that $\operatorname { Im } ( S ) = \operatorname { Im } ( U )$ and that $U U ^ { \mathrm { T } }$ is the matrix of the orthogonal projection onto $\operatorname { Im } ( U )$ in $\mathbb { R } ^ { n }$.
(c) By setting $V = A ^ { \mathrm { T } } U D \in \mathscr { M } _ { p , k } ( \mathbb { R } )$ where $D = \operatorname { diag } \left( 1 / \sqrt { \lambda _ { 1 } } , \ldots , 1 / \sqrt { \lambda _ { k } } \right) \in \mathscr { M } _ { k } ( \mathbb { R } )$, show that $V ^ { \mathrm { T } } V = I _ { k }$ and $\tilde { S } = V \Lambda V ^ { \mathrm { T } }$.