We consider two matrices $A$ and $B$ of $\mathcal{M}_d(\mathbf{R})$ that commute with $[A,B]$. We denote $\mathcal{L} = \operatorname{Vect}(A, B, [A,B])$.
(a) If $M, N \in \mathcal{L}$, show that $[M,N]$ commutes with $M$ and $N$.
(b) Let $G = \{\exp(M) \mid M \in \mathcal{L}\}$. Show that $(G, \times)$ is a group and that the map $$\Phi : \mathbf{H} \rightarrow G, \quad \exp(M_{p,q,r}) \mapsto \exp(pA + qB + r[A,B])$$ is a group homomorphism.

Let $(D_n)_{n \in \mathbf{N}}$ be a sequence of $\mathcal{M}_d(\mathbf{R})$ that converges to $D \in \mathcal{M}_d(\mathbf{R})$. It is therefore bounded: let $\lambda > 0$ be such that for all integers $n \in \mathbf{N}$, $\|D_n\| \leq \lambda$.
(a) Let $k \in \mathbf{N}$. Justify that $\frac{n!}{(n-k)! n^k} \rightarrow 1$ when $n \rightarrow +\infty$ and that if $n \geq k$ (and $n \geq 1$), $$0 \leq 1 - \frac{n!}{(n-k)! n^k} \leq 1$$ Deduce that $$\left(I_d + \frac{D_n}{n}\right)^n - \sum_{k=0}^{n} \frac{1}{k!}(D_n)^k \rightarrow 0 \quad \text{when } n \rightarrow +\infty$$
(b) Show that for all integers $k \geq 1$ and $n \geq 0$, $$\left\|(D_n)^k - D^k\right\| \leq k\lambda^{k-1}\|D_n - D\|$$
(c) Conclude that $\left(I_d + \frac{D_n}{n}\right)^n \rightarrow \exp(D)$ when $n \rightarrow +\infty$.

Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$.
(a) Let $D \in \mathcal{M}_d(\mathbf{R})$ such that $\|D\| \leq 1$. Show that there exists a constant $\mu > 0$ independent of $D$ such that $$\left\|\exp(D) - I_d - D\right\| \leq \mu \|D\|^2$$
(b) Show that there exists a constant $\nu > 0$, and for all $n \geq 1$ a matrix $C_n \in \mathcal{M}_d(\mathbf{R})$, such that $$\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right) = I_d + \frac{A}{n} + \frac{B}{n} + C_n \quad \text{and} \quad \|C_n\| \leq \frac{\nu}{n^2}$$

Let $A$ and $B$ be two arbitrary matrices of $\mathcal{M}_d(\mathbf{R})$. Using the results of questions 8 and 9, deduce that $$\exp(A+B) = \lim_{n \rightarrow +\infty} \left(\exp\left(\frac{A}{n}\right)\exp\left(\frac{B}{n}\right)\right)^n$$

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
If we denote $X_i = \begin{pmatrix} \delta_{1,i} \\ \vdots \\ \delta_{n,i} \end{pmatrix}$ where $\delta_{k,l} = \begin{cases} 1 & \text{if } k = l \\ 0 & \text{if } k \neq l \end{cases}$ and $\mathcal{B}_n = (X_i)_{1 \leqslant i \leqslant n}$ the canonical basis of $E$, what is the matrix of $f$ in $\mathcal{B}_n$?

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$.
V.A.1) Show that there exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal.
This inner product is denoted in the usual way by $\langle u, v \rangle$ or more simply $u \cdot v$ for all $(u, v)$ in $E^2$.
V.A.2) If $u$ and $v$ are represented by the respective column matrices $U$ and $V$ in the basis $\mathcal{B}$, what simple relation exists between $u \cdot v$ and the matrix product ${}^t U V$ (where ${}^t U$ is the transpose of $U$)?

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$. There exists a unique inner product on $E$ for which $\mathcal{B}$ is orthonormal, denoted $\langle u, v \rangle$ or $u \cdot v$.
Let $H$ be a hyperplane of $E$ and $D$ its orthogonal complement. If $(u)$ is a basis of $D$ and if $U$ is the column matrix of $u$ in $\mathcal{B}$, show that $H$ is stable by $f$ if and only if $U$ is an eigenvector of the transpose of $A$.

In this part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$.
Determine thus the stable plane(s) of $f$ when $n = 3$ and $A$ is the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$ considered in IV.F.

Recall why $\mathcal{S}_{n}(\mathbb{R})$ is a real vector space and what is its dimension. Why is the map $s^{\downarrow}$ well-defined on $\mathcal{S}_{n}(\mathbb{R})$?

Is the map $s^{\downarrow}$ linear? Justify your answer.

If $M \in \mathcal{S}_{n}(\mathbb{R})$, express $s^{\downarrow}(-M)$ as a function of the coordinates $\left(m_{1}, \ldots, m_{n}\right)$ of $s^{\downarrow}(M)$.

Let $M = \left(\begin{array}{cc}\lambda & h \\ h & \mu\end{array}\right)$ be a matrix of $\mathcal{S}_{2}(\mathbb{R})$. Calculate $s^{\downarrow}(M)$.

(a) Show that $O _ { n } ( \mathbb { R } )$ is a subgroup of the group $\mathrm { GL } _ { n } ( \mathbb { R } )$ of invertible matrices.
(b) Show that $O _ { n } ( \mathbb { R } )$ is a compact subset of $\mathcal { M } _ { n } ( \mathbb { R } )$.

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum. Show that there exists an orthonormal basis $\left(v_{1}, \ldots, v_{n}\right)$ of $\mathbb{R}^{n}$ such that $$M = \sum_{i=1}^{n} m_{i} v_{i}\, {}^{t}v_{i}$$ Such a decomposition of $M$ will be called in the sequel a spectral resolution of $M$.

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Calculate $$\sup_{\|x\|=1} \langle x, Mx \rangle$$ as a function of the coordinates of $m$. Is this supremum attained? (One may decompose $x$ and $Mx$ on the orthonormal basis $\left(v_{1}, \ldots, v_{n}\right)$ of question 2a).

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Let $j$ be an integer, $1 \leqslant j \leqslant n$. We denote by $\mathcal{V}_{j}$ the vector subspace of $\mathbb{R}^{n}$ spanned by $\left(v_{1}, \ldots, v_{j}\right)$, and by $\mathcal{W}_{j}$ the one spanned by $\left(v_{j}, v_{j+1}, \ldots, v_{n}\right)$. Show the equalities $$\inf_{x \in \mathcal{V}_{j},\, \|x\|=1} \langle x, Mx \rangle = \sup_{x \in \mathcal{W}_{j},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $a_{1} + b_{1} \geqslant c_{1}$.

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
Show that $a_{n} + b_{n} \leqslant c_{n}$.

Let $\mathcal{U}, \mathcal{V}$ and $\mathcal{W}$ be three vector subspaces of $\mathbb{R}^{n}$ such that $$\operatorname{dim} \mathcal{U} + \operatorname{dim} \mathcal{V} + \operatorname{dim} \mathcal{W} > 2n.$$ Show that $\mathcal{U} \cap \mathcal{V} \cap \mathcal{W}$ is not reduced to $\{0\}$.

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$.
By using spectral resolutions of $A$, $B$ and $C$, show that if the strictly positive integers $j$ and $k$ satisfy $j + k \leqslant n + 1$, we have $$c_{j+k-1} \leqslant a_{j} + b_{k}.$$ Deduce that for every integer $j$, $1 \leqslant j \leqslant n$, $$a_{j} + b_{n} \leqslant c_{j}.$$

We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Prove that $a_{11} \leqslant a_{1}$.

Let $j$ and $k$ be non-negative integers such that $1 \leqslant j < k$ and $s_{1} \geqslant s_{2} \geqslant \cdots \geqslant s_{k}$ be real numbers. We define $\mathcal{D}_{j,k} = \left\{\left(t_{1}, \ldots, t_{k}\right) \in [0,1]^{k} \mid t_{1} + \cdots + t_{k} = j\right\}$ and $f$ the function from $\mathcal{D}_{j,k}$ to $\mathbb{R}$ defined by $$f\left(t_{1}, \ldots, t_{k}\right) = \sum_{i=1}^{k} s_{i} t_{i}.$$ Prove that for every $\left(t_{1}, \ldots, t_{k}\right) \in \mathcal{D}_{j,k}$, $$\sum_{i=1}^{j} s_{i} - f\left(t_{1}, \ldots, t_{k}\right) \geqslant \sum_{i=1}^{j} \left(s_{i} - s_{j}\right)\left(1 - t_{i}\right).$$ Deduce that $$\sup_{\mathcal{D}_{j,k}} f = \sum_{i=1}^{j} s_{i}.$$

We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Show that, more generally than in 8a, we have for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} a_{ii} \leqslant \sum_{i=1}^{j} a_{i}.$$

We denote by $a_{ii}$ for $1 \leqslant i \leqslant n$ the diagonal elements of $A \in \mathcal{S}_{n}(\mathbb{R})$ with ordered spectrum $a = s^{\downarrow}(A)$. Deduce that for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} a_{i} = \sup_{\left(x_{1}, \ldots, x_{j}\right) \in \mathcal{R}_{j}} \sum_{i=1}^{j} \langle x_{i}, A x_{i} \rangle,$$ where $\mathcal{R}_{j}$ is the set of orthonormal families of cardinality $j$ in $\mathbb{R}^{n}$.

In this part, we consider two real symmetric matrices $A, B \in \mathcal{S}_{n}(\mathbb{R})$ and their sum $C = A + B$. We denote by $a = s^{\downarrow}(A)$, $b = s^{\downarrow}(B)$ and $c = s^{\downarrow}(C)$. We denote by $\mathcal{R}_{j}$ the set of orthonormal families of cardinality $j$ in $\mathbb{R}^{n}$.
Conclude that for every integer $1 \leqslant j \leqslant n$ $$\sum_{i=1}^{j} c_{i} \leqslant \sum_{i=1}^{j} a_{i} + \sum_{i=1}^{j} b_{i}.$$