Suppose that $M$ and $N$ are two nilpotent matrices that commute. Show that $M N$ and $M + N$ are nilpotent.

Let $H$ be the matrix of the inner product $\phi(P,Q) = \int_0^1 P(t)Q(t)\,\mathrm{d}t$ in the canonical basis of $\mathbb{R}_{n-1}[X]$, with general term $h_{i,j} = \phi(X^i, X^j)$. Show that $H$ belongs to $\mathcal{S}_n(\mathbb{R})$ and that its eigenvalues are strictly positive.

In this part, we denote by $A$ and $B$ two arbitrary matrices in $\mathcal{M}_n(\mathbf{K})$. For every nonzero natural integer $k$, we set $$X_k = \exp\left(\frac{A}{k}\right) \exp\left(\frac{B}{k}\right) \text{ and } Y_k = \exp\left(\frac{A+B}{k}\right).$$
The objective is to prove the relation $$\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}.$$
$\mathbf{8}$ ▷ Verify the relation $$X_k^k - Y_k^k = \sum_{i=0}^{k-1} X_k^i \left(X_k - Y_k\right) Y_k^{k-i-1}$$ Deduce from this the relation $\lim_{k \rightarrow +\infty} \left(e^{\frac{A}{k}} e^{\frac{B}{k}}\right)^k = e^{A+B}$.

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
We denote $$\mathcal{A} = \left\{H \text{ vector subspace of } E \text{ such that } u(H) \subset H \text{ and } F \cap H = \{0_{E}\}\right\}$$ and $$\mathcal{L} = \left\{p \in \mathbf{N}^{*} \mid \exists H \in \mathcal{A} : p = \operatorname{dim}(H)\right\}$$
Prove that $\mathcal{L}$ has a greatest element which we call $r$.

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 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 $\ell \in \mathcal { L } ( E , \mathbb { R } )$, we denote by $\left. \ell \right| _ { F }$ the restriction of $\ell$ to $F$. The restriction map is $r _ { F } : \mathcal { L } ( E , \mathbb { R } ) \rightarrow \mathcal { L } ( F , \mathbb { R } )$, $\ell \mapsto \left. \ell \right| _ { F }$, and $d_{\omega} : E \rightarrow \mathcal{L}(E,\mathbb{R})$, $x \mapsto \omega(x,\cdot)$. The $\omega$-orthogonal is $F ^ { \omega } = \{ x \in E \mid \forall y \in F , \omega ( x , y ) = 0 \}$.
Specify the kernel of $r _ { F } \circ d _ { \omega }$. Deduce that $\operatorname { dim } F ^ { \omega } = \operatorname { dim } E - \operatorname { dim } F$.

Specify the kernel of $r _ { F } \circ d _ { \omega }$. Deduce that $\operatorname { dim } F ^ { \omega } = \operatorname { dim } E - \operatorname { dim } F$.

Suppose that $M , N$ and $M + N$ are nilpotent. By computing $( M + N ) ^ { 2 } - M ^ { 2 } - N ^ { 2 }$, show that $\operatorname { tr } ( M N ) = 0$.

Let $\mathcal{B}_{n}$ be the set of matrices $M$ in $\mathcal{M}_{n}(\mathbf{C})$ such that the sequence $\left(\left\|M^{k}\right\|_{\mathrm{op}}\right)_{k \in \mathbf{N}}$ is bounded. For $M \in \mathcal{B}_{n}$, we set $b(M) = \sup\left\{\left\|M^{k}\right\|_{\mathrm{op}}; k \in \mathbf{N}\right\}$. For $M \in \mathcal{B}_{n}$, we define the function $$\varphi_{M}: z \in \mathbf{C} \backslash \mathbb{D} \longmapsto (|z|-1)\left\|R_{z}(M)\right\|_{\mathrm{op}}.$$
Deduce from the previous question the inequality $$\forall M \in \mathcal{B}_{n}, \quad \forall z \in \mathbf{C} \backslash \mathbb{D}, \quad \varphi_{M}(z) \leq b(M)$$

Show that, if $A$ is nilpotent, that is, if there exists $p \in \mathbb{N}^\star$ such that $A^p = 0_n$, then the spectral radius of $A$ is zero.

In this part, $\mathbf{K} = \mathbf{R}$. For every natural integer $n$, $n \geq 2$, we introduce the set, called the special linear group: $$\mathrm{SL}_n(\mathbf{R}) = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \operatorname{det}(M) = 1 \right\}.$$ If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{9}$ ▷ Determine $\mathcal{A}_G$ when $G = \mathrm{SL}_n(\mathbf{R})$.

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$. We assume that $u$ is diagonalizable. We denote by $\mathcal{B} = (v_{1}, v_{2}, \ldots, v_{n})$ a basis of $E$ formed of eigenvectors of $u$. Let $F$ be a vector subspace of $E$, different from $\{0_{E}\}$ and from $E$.
We denote $$\mathcal{A} = \left\{H \text{ vector subspace of } E \text{ such that } u(H) \subset H \text{ and } F \cap H = \{0_{E}\}\right\}$$ and $$\mathcal{L} = \left\{p \in \mathbf{N}^{*} \mid \exists H \in \mathcal{A} : p = \operatorname{dim}(H)\right\}$$
Prove that $F$ has a complement $G$ in $E$, stable under $u$.

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) 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 $|\Omega_p(e)(e^{\prime})| \leqslant \operatorname{vol}_p(e) \operatorname{vol}_p(e^{\prime})$.

Prove that a matrix $M$ in $\mathcal { M } _ { 2 } ( \mathbb { R } )$ is nilpotent if and only if $\operatorname { det } ( M ) = \operatorname { tr } ( M ) = 0$.

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Prove that $C$ is a closed subset of $\mathcal{M}_{n,1}(\mathbb{R})$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$
$\mathbf{10}$ ▷ If $G = \mathrm{O}_n(\mathbf{R})$, show that $\mathcal{A}_G = \mathcal{A}_n(\mathbf{R})$, the set of antisymmetric matrices.

In this part, $E$ denotes a $\mathbf{C}$-vector space of dimension $n$ and $u$ denotes an endomorphism of $E$.
Assume that every vector subspace of $E$ has a complement in $E$, stable under $u$. Prove that $u$ is diagonalizable. Deduce a characterization of diagonalizable matrices in $M_{n}(\mathbf{C})$.
Hint: one may reason by contradiction and introduce a vector subspace, whose existence one will justify, of dimension $n-1$ and containing the sum of the eigenspaces of $u$.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by
$$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$
where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that
$$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$
where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.

We assume that there exists a symplectic form $\omega$ on $\mathbb { R } ^ { n }$ and we denote by $\Omega \in \mathcal { M } _ { n } ( \mathbb { R } )$ the matrix defined by
$$\Omega = \left( \omega \left( e _ { i } , e _ { j } \right) \right) _ { 1 \leqslant i , j \leqslant n }$$
where $(e _ { 1 } , \ldots , e _ { n })$ denotes the canonical basis of $\mathbb { R } ^ { n }$. Show that
$$\forall ( x , y ) \in \mathbb { R } ^ { n } \times \mathbb { R } ^ { n } , \quad \omega ( x , y ) = X ^ { \top } \Omega Y$$
where $X$ and $Y$ denote the columns of the coordinates of $x$ and $y$ in the canonical basis of $\mathbb { R } ^ { n }$.

Show that the only real nilpotent and symmetric matrix is the zero matrix.

We denote $C = \left\{ U \in \mathcal{M}_{n,1}(\mathbb{R}) \mid U^\top U = 1 \right\}$. Deduce that the application $U \mapsto \left| U^\top A U \right|$ admits a maximum on $C$.

In this part, $\mathbf{K} = \mathbf{R}$. If $G$ is a closed subgroup of $\mathrm{GL}_n(\mathbf{R})$, we introduce its Lie algebra: $$\mathcal{A}_G = \left\{ M \in \mathcal{M}_n(\mathbf{R}) \mid \forall t \in \mathbf{R} \quad e^{tM} \in G \right\}.$$ In questions 11) to 14), $G$ is an arbitrary closed subgroup of $\mathrm{GL}_n(\mathbf{R})$.
$\mathbf{11}$ ▷ Using part 2, show that $\mathcal{A}_G$ is a vector subspace of $\mathcal{M}_n(\mathbf{R})$.

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