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 $$\Omega_p(e)(u) = \det(\operatorname{Gram}(e, u))$$
(a) Verify that the map $(x_1, \ldots, x_p) \mapsto [x_1, \ldots, x_p]$ belongs to $\mathcal{A}_p(\mathbb{R}^p, \mathbb{R})$.
(b) Verify that if $F$ is a vector space over $\mathbb{R}$ and if $f : F \rightarrow \mathbb{R}^p$ is linear, then $g : F^p \rightarrow \mathbb{R}$ defined for $u = (u_1, \ldots, u_p) \in F^p$ by $g(u) = [f(u_1), \ldots, f(u_p)]$ is an element of $\mathcal{A}_p(F, \mathbb{R})$.

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.

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 $t \in ]-\varepsilon_0, \varepsilon_0[$.

Let $A \in S_n^{++}(\mathbf{R})$ and $M \in S_n(\mathbf{R})$. Let $\varepsilon_0 > 0$ be such that for all $t \in ]-\varepsilon_0, \varepsilon_0[, A + tM \in S_n^{++}(\mathrm{R})$. Let $\alpha \in ]-\frac{1}{n}, +\infty[\backslash\{0\}$. We define the application $\varphi_\alpha$ by $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha(t) = \frac{1}{\alpha} \operatorname{det}^{-\alpha}(A + tM).$$ Show that $\varphi_\alpha$ is differentiable on $]-\varepsilon_0, \varepsilon_0[$ and that $$\forall t \in ]-\varepsilon_0, \varepsilon_0[, \quad \varphi_\alpha'(t) = -\operatorname{Tr}\left((A + tM)^{-1}M\right) \operatorname{det}^{-\alpha}(A + tM).$$

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$. Let the application $f _ { A }$ defined on $\mathbf { R }$ by
$$f _ { A } ( t ) = \operatorname { det } ( A + t M )$$
and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Determine $f _ { A } ^ { \prime } ( t )$ for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$.

Let $A \in S _ { n } ^ { + + } ( \mathbf { R } )$ and $M \in S _ { n } ( \mathbf { R } )$, and let $\varepsilon_0 > 0$ be such that for all $t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [ , A + t M \in S _ { n } ^ { + + } ( \mathbf { R } )$. Let $\alpha \in ] - \frac { 1 } { n } , + \infty \left[ \backslash \{ 0 \} \right.$. We define the application $\varphi _ { \alpha }$ by
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \varphi _ { \alpha } ( t ) = \frac { 1 } { \alpha } \operatorname { det } ^ { - \alpha } ( A + t M ) \right.$$
Show that $\varphi _ { \alpha }$ is differentiable on $] - \varepsilon _ { 0 } , \varepsilon _ { 0 } [$ and that
$$\forall t \in ] - \varepsilon _ { 0 } , \varepsilon _ { 0 } \left[ , \quad \varphi _ { \alpha } ^ { \prime } ( t ) = - \operatorname { Tr } \left( ( A + t M ) ^ { - 1 } M \right) \operatorname { det } ^ { - \alpha } ( A + t M ) . \right.$$

Let $J \in M_{n}(\mathbb{C})$ be the matrix defined by
$$J = \left( \begin{array}{ccccc} 0 & 1 & 0 & \ldots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ \vdots & & \ddots & 0 & 1 \\ 1 & 0 & \ldots & 0 & 0 \end{array} \right).$$
Deduce the value of the determinant
$$\left| \begin{array}{ccccc} a_{0} & a_{1} & \ldots & a_{n-2} & a_{n-1} \\ a_{n-1} & a_{0} & \ddots & & a_{n-2} \\ \vdots & \ddots & \ddots & \ddots & \vdots \\ a_{2} & & \ddots & a_{0} & a_{1} \\ a_{1} & a_{2} & \cdots & a_{n-1} & a_{0} \end{array} \right|$$
where $a_{0}, \ldots, a_{n-1}$ are arbitrary complex numbers.

Show that an adjacency matrix of a non-empty graph is never of rank 1.

Show that an adjacency matrix of a graph whose non-isolated vertices form a star-type graph is of rank 2 and represent an example of a graph whose adjacency matrix is of rank 2 and which is not of the previous type.

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_{n}(\mathbb{R})$ $$M_{x} = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right)$$ Deduce that for all $x \in \mathbb{R}$, we have $$\sum_{\sigma \in \mathfrak{S}_{n}} \varepsilon(\sigma) x^{\nu(\sigma)} = (x-1)^{n-1}(x+n-1)$$

Let $n$ be a natural integer with $n \geqslant 2$. For any real number $x$, we consider the following matrix in $\mathscr{M}_n(\mathbb{R})$ $$M_x = \left(\begin{array}{ccccc} x & 1 & \cdots & 1 & 1 \\ 1 & x & \cdots & 1 & 1 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 1 & 1 & \cdots & x & 1 \\ 1 & 1 & \cdots & 1 & x \end{array}\right).$$ Deduce that for all $x \in \mathbb{R}$, we have $$\sum_{\sigma \in \mathfrak{S}_n} \varepsilon(\sigma) x^{\nu(\sigma)} = (x-1)^{n-1}(x+n-1).$$

Let $R \in \mathrm{O}_{d}(\mathbb{R})$. Verify that $\operatorname{det}(R) \in \{-1, +1\}$.