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 $E$).
Deduce from Q37 the existence of an element of rank 1 in $\mathcal{A}$.

Let $f$ be an arithmetic function, $n \in \mathbb{N}^*$ and $g = f * \mu$. We denote $M = \left(m_{ij}\right)$ the matrix of $\mathcal{M}_n(\mathbb{C})$ with general term $m_{ij} = f(i \wedge j)$. We also define the divisor matrix $D = \left(d_{ij}\right)$ by:
$$d_{ij} = \begin{cases} 1 & \text{if } j \text{ divides } i \\ 0 & \text{otherwise} \end{cases}$$
Let $M'$ be the matrix with general term $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Using the result $M = M' D^\top$, deduce that the determinant of $M$ equals
$$\operatorname{det} M = \prod_{k=1}^{n} g(k)$$

Let $\ell \in \llbracket 2, n \rrbracket$ and let $\gamma \in \mathfrak{S}_\ell$ be a cycle of length $\ell$. Show that $\chi_\gamma(X) = X^\ell - 1$.
One may reduce to the case $\gamma = (12\cdots\ell)$ and consider the matrix
$$\Gamma_\ell = \left( \begin{array}{cccccc} 0 & \cdots & \cdots & \cdots & 0 & 1 \\ 1 & 0 & \cdots & \cdots & 0 & 0 \\ 0 & 1 & \ddots & & \vdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots & \vdots \\ \vdots & & \ddots & 1 & 0 & 0 \\ 0 & \cdots & \cdots & 0 & 1 & 0 \end{array} \right) \in \mathcal{M}_\ell(\mathbb{C})$$

In the case $n=1$: Show that a matrix of size $2 \times 2$ is symplectic if and only if its determinant equals 1.
Recall: $J_{1} = \left(\begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array}\right)$, and a matrix $M \in \mathcal{M}_{2}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{1} M = J_{1}$.

Show that the determinant of a symplectic matrix equals either 1 or $-1$.
Recall: A matrix $M \in \mathcal{M}_{2n}(\mathbb{R})$ is symplectic if and only if $M^{\top} J_{n} M = J_{n}$.

Let $f$ be an arithmetic function, $n \in \mathbb{N}^*$ and $g = f * \mu$. We denote $M = \left(m_{ij}\right)$ the matrix of $\mathcal{M}_n(\mathbb{C})$ with general term $m_{ij} = f(i \wedge j)$. We also define the divisor matrix $D = \left(d_{ij}\right)$ by:
$$d_{ij} = \begin{cases} 1 & \text{if } j \text{ divides } i \\ 0 & \text{otherwise} \end{cases}$$
Let $M'$ be the matrix with general term $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Using the result $M = M' D^\top$, deduce that the determinant of $M$ equals
$$\operatorname{det} M = \prod_{k=1}^{n} g(k)$$

We fix a real vector space $E$ of dimension $n$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set
$$H := \operatorname{Vect}(x)^{\perp}, \quad \mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\} \text{ and } \mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$$
We denote by $\pi$ the orthogonal projection of $E$ onto $H$. For $u \in \mathcal{W}$, we denote by $\bar{u}$ the endomorphism of $H$ defined by $\forall z \in H, \bar{u}(z) = \pi(u(z))$. We consider the sets $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$ and $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$.
Show that
$$\operatorname{dim} \mathcal{V} = \operatorname{dim}(\mathcal{V} x) + \operatorname{dim} \mathcal{Z} + \operatorname{dim} \overline{\mathcal{V}}.$$

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the Gram matrix and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Let $Q _ { n } = \left( q _ { i , j } \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$ of $\mathbb { R } _ { n } [ X ]$. Show that $Q _ { n }$ is upper triangular and that $\operatorname { det } Q _ { n } = 1$.

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$ and let $\left( V _ { n } \right) _ { n \in \mathbb { N } }$ be an orthogonal system. Deduce that $\operatorname { det } G _ { n } = \prod _ { i = 0 } ^ { n } \left\| V _ { i } \right\| ^ { 2 }$.

$\mathbf{16}$ ▷ Show that the differential at the point $I_n$ of the application $\det: \mathcal{M}_n(\mathbf{R}) \rightarrow \mathbf{R}$ is the linear form ``trace''.

For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(a) Calculate $\operatorname{vol}_p(b)$ when $b = (b_1, \ldots, b_p)$ is an orthonormal family of vectors of $E$.
(b) Suppose here that $p \geqslant 2$. Let $e = (e_1, \ldots, e_p) \in E^p$. We denote by $\operatorname{pr}$ the orthogonal projection onto the orthogonal of the space spanned by the family $e_2^p = (e_2, \ldots, e_p)$. Show that $\operatorname{vol}_p(e) = \|\operatorname{pr}(e_1)\| \operatorname{vol}_{p-1}(e_2^p)$.
(c) For all free families $e = (e_1, \ldots, e_p) \in E^p$, show that $\operatorname{vol}_p(e) \leqslant \prod_{i=1}^p \|e_i\|$ with equality if and only if $e$ is a family of pairwise orthogonal vectors.

For all $e \in E^p$, we call the $p$-volume of $e$ the quantity $$\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = \left(\operatorname{det}(\operatorname{Gram}(e, e))\right)^{1/2}.$$
(a) Show that if $e \in E^p$ is a free family and if $b \in E^p$ is an orthonormal basis of $\operatorname{Vect}(e)$, then $\operatorname{vol}_p(e) = \left|\operatorname{det}\left(P_b^e\right)\right|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$, i.e. $e_j = \sum_{i=1}^p \left(P_b^e\right)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$.
(b) Show that for all $e, e^{\prime} \in E^p$, we have $\left|\Omega_p(e)(e^{\prime})\right| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$ and $\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = (\det(\operatorname{Gram}(e, e)))^{1/2}$.
(a) Calculate $\operatorname{vol}_p(b)$ when $b = (b_1, \ldots, b_p)$ is an orthonormal family of vectors of $E$.
(b) We assume here that $p \geqslant 2$. Let $e = (e_1, \ldots, e_p) \in E^p$. We denote by $\operatorname{pr}$ the orthogonal projection onto the orthogonal of the space spanned by the family $e_2^p = (e_2, \ldots, e_p)$. Show that $\operatorname{vol}_p(e) = \|\operatorname{pr}(e_1)\| \operatorname{vol}_{p-1}(e_2^p)$.
(c) For all free family $e = (e_1, \ldots, e_p) \in E^p$, show that $\operatorname{vol}_p(e) \leqslant \prod_{i=1}^p \|e_i\|$ with equality if and only if $e$ is a family of vectors that are pairwise orthogonal.

For all $e \in E^p$, we consider $\Omega_p(e) : E^p \rightarrow \mathbb{R}$ defined for all $u \in E^p$ by $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$ and $\operatorname{vol}_p(e) = \sqrt{\Omega_p(e)(e)} = (\det(\operatorname{Gram}(e, e)))^{1/2}$.
(a) Show that if $e \in E^p$ is a free family and if $b \in E^p$ is an orthonormal basis of $\operatorname{Vect}(e)$, then $\operatorname{vol}_p(e) = |\det(P_b^e)|$ where $P_b^e$ is the change of basis matrix from $b$ to $e$ i.e. $e_j = \sum_{i=1}^p (P_b^e)_{ij} b_i$ for all $j \in \llbracket 1, p \rrbracket$.
(b) Show that for all $e, e^{\prime} \in E^p$, we have $|\Omega_p(e)(e^{\prime})| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.

Let $k \in \mathbb{N}^*, (p_1, \ldots, p_k) \in (\mathbb{R}^d)^k$ and $(b_1, \ldots, b_k) \in \mathbb{R}^k$ such that $$A := \left\{x \in \mathbb{R}^d : p_i \cdot x \leqslant b_i, i = 1, \ldots, k\right\}$$ is non-empty. Show that $A$ is convex and closed. Let $x \in A$, let $I(x) := \{i \in \{1, \ldots, k\} : p_i \cdot x = b_i\}$, show that $$x \in \operatorname{Ext}(A) \Longleftrightarrow \operatorname{rank}\left(\{p_i, i \in I(x)\}\right) = d$$ deduce that $\operatorname{Ext}(A)$ is a finite set (possibly empty) whose cardinality is at most $2^k$.

Let $k \in \mathbb{N}^*, (p_1, \ldots, p_k) \in (\mathbb{R}^d)^k$ and $(b_1, \ldots, b_k) \in \mathbb{R}^k$ such that $$A := \left\{x \in \mathbb{R}^d : p_i \cdot x \leqslant b_i, i = 1, \ldots, k\right\}$$ is non-empty. Show that $A$ is convex and closed. Let $x \in A$, let $I(x) := \left\{i \in \{1, \ldots, k\} : p_i \cdot x = b_i\right\}$, show that $$x \in \operatorname{Ext}(A) \Longleftrightarrow \operatorname{rank}\left(\{p_i, i \in I(x)\}\right) = d$$ deduce that $\operatorname{Ext}(A)$ is a finite set (possibly empty) whose cardinality is at most $2^k$.

We have $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$ and for every $M \in \mathcal{C}_J$, $\det(M) \geq 0$. Deduce that, for every matrix $M$ in $\operatorname { OSp } _ { n } ( \mathbb { R } ) , \operatorname { det } ( M ) = 1$.

Using the polar decomposition $M = OS$ where $O \in \operatorname{OSp}_n(\mathbb{R})$ and $S \in \mathrm{Sp}_n(\mathbb{R})$ is symmetric with strictly positive eigenvalues, conclude that the determinant of the matrix $M \in \mathrm{Sp}_n(\mathbb{R})$ is equal to 1.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ with associated matrix $\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$ satisfying $\omega(x,y) = X^{\top} \Omega Y$ for all $x,y \in \mathbb{R}^n$. Deduce that $\Omega$ is antisymmetric and invertible.

Given that any symplectic form on $\mathbb{R}^n$ has an associated matrix $\Omega$ that is antisymmetric and invertible, conclude that the integer $n$ is even.

Given that $\operatorname { OSp } _ { n } ( \mathbb { R } ) \subset \mathcal { C } _ { J }$ and that for every matrix $M \in \mathcal { C } _ { J }$, $\det(M) \geq 0$, deduce that, for every matrix $M$ in $\operatorname { OSp } _ { n } ( \mathbb { R } ) , \operatorname { det } ( M ) = 1$.

Using the polar decomposition $M = OS$ where $O \in \operatorname{OSp}_n(\mathbb{R})$ and $S$ is a symmetric symplectic matrix with strictly positive eigenvalues, conclude that the determinant of the matrix $M \in \mathrm{Sp}_n(\mathbb{R})$ is equal to 1.

Determine the trace and the determinant of a nilpotent matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$.

Let $Q$ be a delta endomorphism, let $(q_n)_{n \in \mathbb{N}}$ be the sequence of polynomials associated with $Q$, and let $n$ be a natural number. The family $(q_0, q_1, \ldots, q_n)$ is a basis of $\mathbb{K}_n[X]$.
According to question 23, $Q$ induces an endomorphism of $\mathbb{K}_n[X]$ denoted $Q_n$. Give its matrix in the previous basis. Deduce its trace, its determinant and its characteristic polynomial.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let the application $f_A$ defined on $\mathbf{R}$ by $$f_A(t) = \operatorname{det}(A + tM).$$ Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Determine $f_A'(t)$ for all $t \in ]-\varepsilon_0, \varepsilon_0[$.