Show that if $i$ and $j$ are in $\llbracket -n+1, n-1 \rrbracket$, if $A \in \Delta_i$ and $B \in \Delta_j$, then $AB \in \Delta_{i+j}$.

Deduce that if $A \in H_i$ and $B \in H_j$, then $AB \in H_{i+j}$.

We call the commutant of $f$ the set $\mathcal{C}(f) = \{g \in \mathcal{L}(E) \mid f \circ g = g \circ f\}$. Show that $\mathcal{C}(f)$ is a subalgebra of $\mathcal{L}(E)$.

Are the subsets $S_{2}(\mathbb{K})$ and $A_{2}(\mathbb{K})$ subalgebras of $\mathcal{M}_{2}(\mathbb{K})$?

Suppose $n \geqslant 3$. Are the subsets $\mathrm{S}_{n}(\mathbb{K})$ and $\mathrm{A}_{n}(\mathbb{K})$ subalgebras of $\mathcal{M}_{n}(\mathbb{K})$?

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 $m_{ij}' = \begin{cases} g(j) & \text{if } j \text{ divides } i, \\ 0 & \text{otherwise.} \end{cases}$
Show that $M = M' D^\top$, where $D^\top$ is the transpose of $D$.

3a. We are given $q \in \mathbb { Q }$, $n \in \mathbb { N } ^ { \star }$ and a matrix $A \in S _ { n } ( \mathbb { Q } )$ such that $A ^ { 2 } = q I _ { n }$. Construct a matrix $B \in S _ { 2 n } ( \mathbb { Q } )$ commuting with the matrix $\left( \begin{array} { c c } A & 0 \\ 0 & A \end{array} \right)$ and such that $B ^ { 2 } = ( q + 1 ) I _ { 2 n }$.
3b. Show that for all $d \geqslant 1$, there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { k } ^ { 2 } = k I _ { n }$ for all integers $1 \leqslant k \leqslant d$.
3c. Let $d \geqslant 1$ be an integer. Deduce that if $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q }$, $q _ { i } > 0$, then there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { i } ^ { 2 } = q _ { i } I _ { n }$ for all $1 \leqslant i \leqslant d$.

3a. We are given $q \in \mathbb { Q } , n \in \mathbb { N } ^ { \star }$ and a matrix $A \in S _ { n } ( \mathbb { Q } )$ such that $A ^ { 2 } = q I _ { n }$. Construct a matrix $B \in S _ { 2 n } ( \mathbb { Q } )$ commuting with the matrix $\left( \begin{array} { c c } A & 0 \\ 0 & A \end{array} \right)$ and such that $B ^ { 2 } = ( q + 1 ) I _ { 2 n }$.
3b. Show that for all $d \geqslant 1$, there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { k } ^ { 2 } = k I _ { n }$ for all integers $1 \leqslant k \leqslant d$.
3c. Let $d \geqslant 1$ be an integer. Deduce that if $q _ { 1 } , \ldots , q _ { d } \in \mathbb { Q } , q _ { i } > 0$, then there exist $n \in \mathbb { N } ^ { \star }$ and matrices $M _ { 1 } , \ldots , M _ { d } \in S _ { n } ( \mathbb { Q } )$ that commute pairwise and such that $M _ { i } ^ { 2 } = q _ { i } I _ { n }$ for all $1 \leqslant i \leqslant d$.

We consider a non-zero totally real number $z$. We write $Z ( X ) = X ^ { d } - \left( a _ { d - 1 } X ^ { d - 1 } + \cdots + a _ { 1 } X + a _ { 0 } \right)$ with $d$ minimal and $a_i \in \mathbb{Q}$. We consider the matrix $S$ of size $d \times d$ whose coefficient $( i , j )$ equals $t \left( z ^ { i + j } \right)$, and $P \in \mathrm{GL}_d(\mathbb{Q})$, $q_1, \ldots, q_d \in \mathbb{Q}$, $q_i > 0$ such that $S = P^T \cdot \operatorname{Diag}(q_1, \ldots, q_d) \cdot P$. We set: $$M = \left( \begin{array} { c c c c c } 0 & 0 & \cdots & 0 & a _ { 0 } \\ 1 & 0 & \ddots & 0 & a _ { 1 } \\ 0 & \ddots & \ddots & \vdots & \vdots \\ \vdots & \ddots & \ddots & 0 & a _ { d - 2 } \\ 0 & \cdots & 0 & 1 & a _ { d - 1 } \end{array} \right)$$
17a. Verify that the matrix $S M$ is symmetric.
17b. Deduce that the matrix $R M R ^ { - 1 }$ is symmetric where $R = \operatorname { Diag } \left( \sqrt { q _ { 1 } } , \ldots , \sqrt { q _ { d } } \right) \cdot P$.

Let $n \in \mathbf{N}$ and $P : \mathbf{R} \rightarrow \mathbf{R}$ defined by $P(x) = \sum_{k=0}^{n} a_k x^k$ where $a_k \geq 0$ for all $k \in \llbracket 0, n \rrbracket$ a polynomial with non-negative coefficients.
(a) Verify that $P[A] = \sum_{k=0}^{n} a_k A^{(k)}$ for all matrices $A \in \mathcal{M}_p(\mathbf{R})$.
(b) Show that if $A \in \operatorname{Sym}^+(p)$ then $P[A] \in \operatorname{Sym}^+(p)$.

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Let $k \in \mathbf{N}^{*}$. Show that there exists a unique family $(f_{0}^{(k)}, \ldots, f_{k}^{(k)})$ of endomorphisms of $E$ such that
$$\forall t \in \mathbf{R}, (u + tv)^{k} = \sum_{i=0}^{k} t^{i} f_{i}^{(k)}$$
Show in particular that $f_{0}^{(k)} = u^{k}$ and $f_{1}^{(k)} = \sum_{i=0}^{k-1} u^{i} v u^{k-1-i}$.

We consider an $\mathbf{R}$-vector space $E$ of dimension $n > 0$. Let $\mathcal{V}$ be a nilpotent vector subspace of $\mathcal{L}(E)$ containing a non-zero element, with generic nilindex $p := \max_{u \in \mathcal{V}} \nu(u)$. In questions 8 to 11, we are given two arbitrary elements $u$ and $v$ of $\mathcal{V}$.
Show that $\sum_{i=0}^{p-1} u^{i} v u^{p-1-i} = 0$.

For all $n \in \mathbb { N }$, let $G _ { n } = \left( \left( X ^ { i - 1 } \mid X ^ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, let $G _ { n } ^ { \prime } = \left( \left( V _ { i - 1 } \mid V _ { j - 1 } \right) \right) _ { 1 \leqslant i , j \leqslant n + 1 }$, and let $Q _ { n }$ be the matrix of the family $\left( V _ { 0 } , V _ { 1 } , \ldots , V _ { n } \right)$ in the basis $\left( 1 , X , \ldots , X ^ { n } \right)$. Show that $Q _ { n } ^ { \top } G _ { n } Q _ { n } = G _ { n } ^ { \prime }$, where $Q _ { n } ^ { \top }$ is the transpose of the matrix $Q _ { n }$.

We consider the graph $G$ represented in Figure 2. We recall that, when the point is on one of the vertices of the graph, it has the same probability of going to each of the vertices to which it is connected. We assume that initially, the point is on vertex 1, so that $P ^ { ( 0 ) } = ( 1,0,0,0,0,0,0,0 )$. We denote $S _ { 1 } = \{ 1,3,6,8 \}$ and $S _ { 2 } = \{ 2,4,5,7 \}$.
Give the transition matrix $T$ of this graph and calculate $$( 1,1,1,1,1,1,1,1 ) T .$$

Let $M \in \mathcal { M } _ { n } ( \mathbb { R } )$ and $N \in \mathcal { M } _ { n } ( \mathbb { R } )$ be two stochastic matrices, $X \in \mathbb { R } ^ { n }$ a probability distribution and $\alpha \in [ 0,1 ]$. Show that $M N$ is a stochastic matrix.

Let $M \in \mathcal { M } _ { n } ( \mathbb { R } )$ and $N \in \mathcal { M } _ { n } ( \mathbb { R } )$ be two stochastic matrices, $X \in \mathbb { R } ^ { n }$ a probability distribution and $\alpha \in [ 0,1 ]$. Show that $\alpha M + ( 1 - \alpha ) N$ is a stochastic matrix.

We are given two matrices $A$ and $B$ in $\mathcal{M}_n(\mathbf{K})$. We assume that $A$ and $B$ commute.
$\mathbf{1}$ ▷ Show that the matrices $A$ and $e^{B}$ commute.

$\mathbf{3}$ ▷ Conversely, suppose the relation $\forall t \in \mathbf{R} \quad e^{t(A+B)} = e^{tA} e^{tB}$ is satisfied. By differentiating this relation twice with respect to the real variable $t$, show that the matrices $A$ and $B$ commute.

Let $e = (e_1, \ldots, e_d)$ be an orthonormal basis of $E$, $p$ an integer such that $1 \leqslant p \leqslant d$ and $\mathcal{I}_p = \{\alpha = (i_1, \ldots, i_p) \in \mathbb{N}^p \mid 1 \leqslant i_1 < \cdots < i_p \leqslant d\}$. For all $\omega$ and $\omega^{\prime}$ elements of $\mathcal{A}_p(E, \mathbb{R})$: $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha})$$
We consider $u, v \in E^p$. Show that $$\Omega_p(u)(v) = \langle\Omega_p(u), \Omega_p(v)\rangle$$

Let $A$ and $B$ be two matrices in $\mathcal { M } _ { n } ( \mathbb { R } )$ such that
$$\forall ( X , Y ) \in \left( \mathcal { M } _ { n , 1 } ( \mathbb { R } ) \right) ^ { 2 } , \quad X ^ { \top } A Y = X ^ { \top } B Y .$$
Show that $A = B$.

Prove that the application $$\begin{array} { | c l l } \mathcal { M } _ { n } ( \mathbb { R } ) & \rightarrow & \mathbb { R } \\ M & \mapsto & \operatorname { tr } ( M ) \end{array}$$ is a linear form and that $$\forall ( A , B ) \in \left( \mathcal { M } _ { n } ( \mathbb { R } ) \right) ^ { 2 } , \quad \operatorname { tr } ( A B ) = \operatorname { tr } ( B A ) .$$

Show that the application $$\begin{array} { | c c c } \left( \mathcal { M } _ { n } ( \mathbb { R } ) \right) ^ { 2 } & \rightarrow & \mathbb { R } \\ ( A , B ) & \mapsto & \operatorname { tr } \left( A ^ { \top } B \right) \end{array}$$ is an inner product on $\mathcal { M } _ { n } ( \mathbb { R } )$.

Deduce that if $A$ is a matrix in $\mathcal { M } _ { n } ( \mathbb { R } )$ satisfying $A ^ { \top } A = 0$ then $A = 0$.

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

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