We denote $\alpha > -1/2$, $F_n$ the vector subspace of $E$ of polynomial functions of degree less than or equal to $n$ (where $n \in \mathbb{N}$), $$\varphi_\alpha(y) : t \mapsto \left(1-t^2\right)y''(t) - (2\alpha+1)t\,y'(t)$$ and $$S_\alpha(f,g) = \int_{-1}^{1} f(t)g(t)\left(1-t^2\right)^{\alpha - \frac{1}{2}} \mathrm{~d}t$$ Justify that two eigenvectors of $\varphi_\alpha$ of distinct degrees are orthogonal (with respect to $S_\alpha$).

In the usual Euclidean space $\mathbb { R } ^ { 3 }$, show that, for all vectors $u$ and $v$ of $\mathbb { R } ^ { 3 }$ of the same norm, there exists a rotation $r$ such that $r ( u ) = v$.

Write, in Maple or Mathematica language, a function (or procedure) rotation, with parameters $U$ and $V$, returning:

• False if $U$ and $V$ do not have the same norm;
• a matrix $R$ of $S O ( 3 )$ such that $R U = V$ if $U$ and $V$ have the same norm.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero.
Deduce from question III.E.1 that there exists an element $L _ { 1 }$ of $G$ such that: $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ where $\alpha$ is a strictly positive real number that we will specify, $\lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ are real numbers that we will not seek to determine.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix coefficients $\alpha , \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ such that $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ Let $v _ { 2 } = \left( \begin{array} { l } \mu _ { 1 } \\ \mu _ { 2 } \\ \mu _ { 3 } \end{array} \right)$ and $v _ { 3 } = \left( \begin{array} { l } \nu _ { 1 } \\ \nu _ { 2 } \\ \nu _ { 3 } \end{array} \right)$. Show that $v _ { 2 }$ and $v _ { 3 }$ are two unit vectors orthogonal to each other in $\mathbb { R } ^ { 3 }$ equipped with its usual Euclidean structure.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix coefficients $\alpha , \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } , \mu _ { 1 } , \mu _ { 2 } , \mu _ { 3 } , \nu _ { 1 } , \nu _ { 2 }$ and $\nu _ { 3 }$ such that $$L _ { 1 } L = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \ell _ { 1,2 } & \ell _ { 1,3 } & \ell _ { 1,4 } \\ \alpha & \lambda _ { 1 } & \lambda _ { 2 } & \lambda _ { 3 } \\ 0 & \mu _ { 1 } & \mu _ { 2 } & \mu _ { 3 } \\ 0 & \nu _ { 1 } & \nu _ { 2 } & \nu _ { 3 } \end{array} \right)$$ Let $R _ { 2 } \in S O ( 3 )$. We set $L _ { 2 } = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R _ { 2 } \end{array} \right) \in G$. Show that we can choose $R _ { 2 }$ such that $$L _ { 1 } L L _ { 2 } = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \beta _ { 1 } & \beta _ { 2 } & \beta _ { 3 } \\ \alpha & \delta _ { 1 } & \delta _ { 2 } & \delta _ { 3 } \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$ where $\beta _ { 1 } , \beta _ { 2 } , \beta _ { 3 } , \delta _ { 1 } , \delta _ { 2 }$ and $\delta _ { 3 }$ are real numbers that we will not seek to determine.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Let $L = \left( \ell _ { i , j } \right) _ { 1 \leqslant i , j \leqslant 4 } \in O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ and $a = \left( \begin{array} { c } \ell _ { 2,1 } \\ \ell _ { 3,1 } \\ \ell _ { 4,1 } \end{array} \right)$. We assume that the vector $a$ is non-zero. We fix $L_1 \in G$ and $R_2 \in SO(3)$ such that $$L _ { 1 } L L _ { 2 } = \left( \begin{array} { c c c c } \ell _ { 1,1 } & \beta _ { 1 } & \beta _ { 2 } & \beta _ { 3 } \\ \alpha & \delta _ { 1 } & \delta _ { 2 } & \delta _ { 3 } \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right)$$ where $L_2 = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R _ { 2 } \end{array} \right)$.
Show that the real numbers $\beta _ { 2 } , \beta _ { 3 } , \delta _ { 2 }$ and $\delta _ { 3 }$ are zero.

We set $$G = \left\{ \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) , R \in S O ( 3 ) \right\}$$ Deduce that every matrix $L$ of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$ can be written in the form of a product of the type $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ where $R$ and $R ^ { \prime }$ are two elements of $S O ( 3 )$ and $\gamma$ is a real number.

Write, in Maple or Mathematica language, a function or procedure allowing to obtain such a decomposition $$L = \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R \end{array} \right) \left( \begin{array} { c c c c } \operatorname { ch } \gamma & \operatorname { sh } \gamma & 0 & 0 \\ \operatorname { sh } \gamma & \operatorname { ch } \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) \left( \begin{array} { c c } 1 & 0 _ { 1,3 } \\ 0 _ { 3,1 } & R ^ { \prime } \end{array} \right)$$ of a matrix of $O ^ { + } ( 1,3 ) \cap \tilde { O } ( 1,3 )$. You may use the rotation function written previously.

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$.
(a) Let $s _ { 1 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 1 }$. Show that the matrix of $s _ { 1 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } 1 & 1 \\ 0 & - 1 \end{array} \right)$.
(b) Let $s _ { 2 } : E \longrightarrow E$ be the orthogonal reflection with respect to the line $\mathbb { R } \omega _ { 2 }$. Show that the matrix of $s _ { 2 }$ in the basis $\mathcal { C }$ is $\left( \begin{array} { c c } - 1 & 0 \\ 1 & 1 \end{array} \right)$.

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$. We denote by $H ^ { + }$ the subset of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$ such that $m _ { 1 } \geqslant m _ { 2 } \geqslant m _ { 3 }$. We consider the application $$\begin{array} { c c c c } \varphi : & H & \longrightarrow & E \\ & \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) & \longmapsto & \left( m _ { 1 } - m _ { 2 } \right) \omega _ { 1 } + \left( m _ { 2 } - m _ { 3 } \right) \omega _ { 2 } \end{array}$$ (a) Show that $\varphi$ is a linear isomorphism. Describe $\varphi \left( H ^ { + } \right)$.
(b) Show that, for all $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$, we have $$s _ { 1 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right) \quad \text { and } \quad s _ { 2 } \circ \varphi \left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) = \varphi \left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right) .$$ (c) Let $\widehat { \lambda } = \left( \lambda _ { 1 } , \lambda _ { 2 } , \lambda _ { 3 } \right) \in H$ such that $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$. We denote by $\mathcal { Q } _ { \widehat { \lambda } }$ the set of $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H ^ { + }$ such that $m _ { 1 } \leqslant \lambda _ { 1 }$ and $m _ { 1 } + m _ { 2 } \leqslant \lambda _ { 1 } + \lambda _ { 2 }$. Show that $\varphi \left( \mathcal { Q } _ { \widehat { \lambda } } \right)$ is a quadrilateral whose vertices will be described.

We denote by $E$ the vector space $\mathbb { R } ^ { 2 }$ equipped with the standard inner product and the canonical basis $\mathcal { B } = \left\{ e _ { 1 } , e _ { 2 } \right\}$. We define a basis $\mathcal { C } = \left\{ \omega _ { 1 } , \omega _ { 2 } \right\}$ of $E$ by $\omega _ { 1 } = e _ { 1 }$ and $\omega _ { 2 } = \frac { 1 } { 2 } \left( e _ { 1 } + \sqrt { 3 } e _ { 2 } \right)$. Let $s_1$ be the orthogonal reflection with respect to $\mathbb{R}\omega_1$ and $s_2$ the orthogonal reflection with respect to $\mathbb{R}\omega_2$. Let $H$ be the set of vectors $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathbb { R } ^ { 3 }$ such that $m _ { 1 } + m _ { 2 } + m _ { 3 } = 0$, $H^+$ the subset with $m_1 \geqslant m_2 \geqslant m_3$, and $\varphi : H \to E$ defined by $\varphi(m_1,m_2,m_3) = (m_1-m_2)\omega_1 + (m_2-m_3)\omega_2$. We consider the application $$\begin{array} { r c c c } \operatorname { Diag } _ { n } : & S _ { n } ( \mathbb { R } ) & \longrightarrow & \mathbb { R } ^ { n } \\ M = \left( m _ { i j } \right) & \longmapsto & \left( m _ { 11 } , m _ { 22 } , \ldots , m _ { n n } \right) \end{array}$$ and $\mathcal { D } _ { M } = \left\{ \operatorname { Diag } _ { n } \left( U M U ^ { - 1 } \right) , \text { for } U \text { ranging over } O _ { n } ( \mathbb { R } ) \right\}$. Let $M \in S _ { 3 } ( \mathbb { R } )$ be a matrix with zero trace. We denote $\operatorname { Sp } ( M ) = \left( \lambda _ { 1 } \geqslant \lambda _ { 2 } \geqslant \lambda _ { 3 } \right)$. We propose to describe $\varphi \left( \mathcal { D } _ { M } \right)$.
(a) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in H$. Let $\sigma$ be a permutation of $\{ 1,2,3 \}$. Show that $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$ if and only if $\left( m _ { \sigma ( 1 ) } , m _ { \sigma ( 2 ) } , m _ { \sigma ( 3 ) } \right) \in \mathcal { D } _ { M }$.
(b) Using question 13, show that the intersection $H ^ { + } \cap \mathcal { D } _ { M }$ is contained in $\mathcal { Q } _ { \widehat { \lambda } }$.
(c) Let $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right) \in \mathcal { D } _ { M }$. Show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 2 } , m _ { 1 } , m _ { 3 } \right)$ is contained in $\mathcal { D } _ { M }$. One may use question 12. Similarly, show that the segment of $H$ whose vertices are $\left( m _ { 1 } , m _ { 2 } , m _ { 3 } \right)$ and $\left( m _ { 1 } , m _ { 3 } , m _ { 2 } \right)$ is contained in $\mathcal { D } _ { M }$.
(d) Show that $\mathcal { D } _ { M }$ contains $\mathcal { Q } _ { \widehat { \lambda } }$.
(e) Show that if $\lambda _ { 1 } > \lambda _ { 2 } > \lambda _ { 3 }$ then $\varphi \left( \mathcal { D } _ { M } \right)$ is a hexagon, whose vertices will be determined.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$, and suppose there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$.
Give the matrix of $u$ in a basis adapted to the decomposition $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$.
Show that there exist a partition $\sigma = \left(\alpha_1, \ldots, \alpha_k\right)$ of $n$ and a basis $\mathcal{B}$ of $E$ in which the matrix of $u$ is equal to the matrix $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$.

Let $\alpha$ be a non-zero natural integer. Calculate the rank of $J_\alpha^j$ for every natural integer $j$. Deduce that $J_\alpha$ is nilpotent and specify its nilpotency index.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$ with $\alpha_1 \geqslant \cdots \geqslant \alpha_k$.
Deduce the value of $\alpha_1$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For $j \in \mathbb{N}$, we denote $\Lambda_j = \left\{i \in \llbracket 1, k \rrbracket \mid \alpha_i \geqslant j\right\}$. Prove that $\operatorname{rg}\left(N_\sigma^j\right) = \sum_{i \in \Lambda_j} \left(\alpha_i - j\right)$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Prove that, for every $j \in \mathbb{N}^*$, the integer $d_j = \operatorname{rg}\left(u^{j-1}\right) - \operatorname{rg}\left(u^j\right)$ is equal to the number of blocks $J_{\alpha_i}$ whose size $\alpha_i$ is greater than or equal to $j$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Give the value of the integer $k$, the number of blocks $J_{\alpha_i}$ appearing in $N_\sigma$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
For every integer $j$ between 1 and $n$, express the number of blocks $J_{\alpha_i}$ of size exactly equal to $j$.

Let $u$ be an endomorphism of $E$ nilpotent of index $p$ and of rank $r$. Suppose the matrix of $u$ in some basis $\mathcal{B}$ is $N_\sigma = \operatorname{diag}\left(J_{\alpha_1}, \ldots, J_{\alpha_k}\right)$ where $\sigma = (\alpha_1, \ldots, \alpha_k)$ is a partition of $n$.
Assume that there exist a partition $\sigma'$ of the integer $n$ and a basis $\mathcal{B}'$ of $E$ such that the matrix of $u$ in $\mathcal{B}'$ is equal to $N_{\sigma'}$. Show that $\sigma = \sigma'$.

What is the maximum cardinality of a set of nilpotent matrices, all of the same size $n$, such that there are no two similar matrices in this set?

To every $p \in \mathbb{R}[X]$, we associate the function $J(p) = Jp$ from $\mathbb{R}$ to $\mathbb{R}$ defined by $$\forall x \in \mathbb{R}, \quad J(p)(x) = Jp(x) = \int_x^{x+1} p(t)\,\mathrm{d}t$$
Show that $J$ is an endomorphism of $\mathbb{R}[X]$.

To every $p \in \mathbb{K}[X]$, we associate the function $L(p) = Lp$ from $\mathbb{K}$ to $\mathbb{K}$ defined by $$\forall x \in \mathbb{K}, \quad L(p)(x) = Lp(x) = -\int_0^{+\infty} \mathrm{e}^{-t} p'(x+t)\,\mathrm{d}t$$
Show that $L$ is an endomorphism of $\mathbb{K}[X]$. Is it invertible?

Let $a \in \mathbb{K}$. Verify that the endomorphisms $I$ and $D$ are shift-invariant, as well as the endomorphisms $E_a$, $J$ and $L$ defined in part I. Are they delta endomorphisms?
Recall: $T$ is shift-invariant if for all $a \in \mathbb{K}$, $E_a \circ T = T \circ E_a$. $T$ is a delta endomorphism if $T$ is shift-invariant and $TX \in \mathbb{K}^*$.