Calculate the exponential of the matrix $M_{p,q,r}$, where $$M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix}.$$
(a) Show that we define a group law $*$ on $\mathbf{L}$ by setting for $M, N \in \mathbf{L}$: $$M * N = M + N + \frac{1}{2}[M, N]$$ Explicitly determine the inverse of $M_{p,q,r}$.
(b) Determine the matrices $M_{p,q,r} \in \mathbf{L}$ that commute with all elements of $\mathbf{L}$ for the law $*$. Is $(\mathbf{L}, *)$ commutative?

We consider the set of square matrices of size 3 that are strictly upper triangular: $$\mathbf{L} = \left\{ M_{p,q,r} \mid (p,q,r) \in \mathbf{R}^3 \right\} \quad \text{where} \quad M_{p,q,r} = \begin{pmatrix} 0 & p & r \\ 0 & 0 & q \\ 0 & 0 & 0 \end{pmatrix},$$ and the group law $M * N = M + N + \frac{1}{2}[M,N]$ on $\mathbf{L}$.
Show that for all matrices $M, N \in \mathbf{L}$, we have: $$(\exp M) \times (\exp N) = \exp(M * N)$$

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 $\mathcal{B}_n$?

Determine a pair $(A, \vec{b})$ in $\mathrm{SO}(2) \times \mathbb{R}^2$ such that $M(A, \vec{b}) = I_3$.

Let $(A, \vec{b})$ and $(A^{\prime}, \vec{b}^{\prime})$ be in $\mathrm{SO}(2) \times \mathbb{R}^2$. Show that $M(A, \vec{b}) M\left(A^{\prime}, \vec{b}^{\prime}\right) = M\left(A A^{\prime}, A \vec{b}^{\prime} + \vec{b}\right)$.

Show that the elements of $G$ are invertible and explicitly determine the inverse of $M(A, \vec{b})$.

Prove that $G$ is a subgroup of $\mathrm{GL}_3(\mathbb{R})$.

Is the application $\Phi : \left\{ \begin{array}{cll} G & \rightarrow & \mathbb{R}^2 \\ M(A, \vec{b}) & \mapsto & \vec{b} \end{array} \right.$ surjective? Is it injective?

Let $m \in \mathbb{N}$. We denote by $\mathcal{H}_m$ the vector subspace of harmonic polynomials of degree less than or equal to $m$. Determine the dimension of $\mathcal{H}_m$.

We work on $\mathbb{R}^n$ for a natural integer $n \geqslant 3$. Let $m \in \mathbb{N}^*$.
Show that the set $$\left\{ (i_1, i_2, \ldots, i_n) \in \mathbb{N}^n \mid i_1 + i_2 + \cdots + i_n = m \right\}$$ has cardinality $\dbinom{n+m-1}{m}$. Deduce the dimension of $\mathcal{P}_m$.

We work on $\mathbb{R}^n$ for a natural integer $n \geqslant 3$. We admit that the Dirichlet problem on the unit ball of $\mathbb{R}^n$, associated with a continuous function defined on the unit sphere $S_n(0,1)$, admits a unique solution. Let $m \in \mathbb{N}^*$.
Determine the dimension of $\mathcal{H}_m$ as a function of $m$ and $n$.

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 $s^{\downarrow}(M)$.

We denote by $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ the set of $n \times n$ symmetric matrices whose eigenvalues are all simple. Let $M \in \mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Determine a real $r > 0$ such that the open ball of $\mathcal{S}_{n}(\mathbb{R})$ centered at $M$ with radius $r$ is included in $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$. Deduce that $\mathcal{S}_{n}^{\dagger}(\mathbb{R})$ is an open set of $\mathcal{S}_{n}(\mathbb{R})$.

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 study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $a_{1} \geqslant a_{2}$ and $b_{1} \geqslant b_{2}$, four real numbers satisfying the relation $$a_{1} - a_{2} \geqslant b_{1} - b_{2}.$$ We seek to identify the set $$\Sigma = \left\{s^{\downarrow}(A + B) \mid A \in S\left(a_{1}, a_{2}\right),\, B \in S\left(b_{1}, b_{2}\right)\right\}.$$
Show that $\Sigma$ is included in a line segment $L$ of length $\sqrt{2}\left(b_{1} - b_{2}\right)$, and determine its endpoints. One may first study the case where $A$ and $B$ are diagonal.

In this part, we study the case $n = 2$. For two real numbers $u$ and $v$ such that $u \geqslant v$, we denote: $$S(u, v) = \left\{M \in \mathcal{S}_{2}(\mathbb{R}) \mid s^{\downarrow}(M) = (u, v)\right\}.$$ We fix $b_{1} \geqslant b_{2}$.
Determine a continuous function defined on $[-\pi, \pi]$ whose image equals $S\left(b_{1}, b_{2}\right)$.

Justify that $\mathcal{X}_n$ is a finite set and determine its cardinality.

Prove that for all $M \in \mathcal{Y}_n$, $\operatorname{det}(M) \leqslant n!$ and that there is no equality.

Let $M \in \mathcal{Y}_n$ and $\lambda$ a complex eigenvalue of $M$. Prove that $|\lambda| \leqslant n$ and give an explicit example where equality holds.

List the elements of $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$. Specify (by justifying) which ones are diagonalizable over $\mathbb{R}$.

Prove that $\mathcal{X}_2' = \mathcal{X}_2 \cap \mathrm{GL}_2(\mathbb{R})$ generates the vector space $\mathcal{M}_2$. For $n \geqslant 2$, does $\mathcal{X}_n'$ generate the vector space $\mathcal{M}_n(\mathbb{R})$?