Let $M \in M_{n}(\mathbb{C})$. Show that $M$ is diagonalizable if and only if for every polynomial $P(X) \in \mathbb{C}[X]$ such that $P(M)$ is nilpotent, $P(M) = 0$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$. We fix $x \in E \backslash \{0\}$. By considering $\Phi : f \mapsto f(x)$, linear application from $V$ to $E$, show that $\operatorname{dim}(V) \leqslant n$. Thus $1 \leqslant d_{n} \leqslant n$.

Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of dimension $d \geqslant 1$. Show that there exists a vector subspace $W$ of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of the same dimension $d$, and containing $\operatorname{Id}_{E}$.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$.
For $k \in \{ 1,2 \}$, we denote by $e _ { k }$ the element of $\mathcal { E } ^ { * }$ which associates to every matrix $\left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right)$, where $D = \left( \begin{array} { c c } d _ { 1 } & 0 \\ 0 & d _ { 2 } \end{array} \right) \in \mathcal { D } ( 2 , \mathbb { R } )$, the coefficient $d _ { k }$.
a) Verify that $(e _ { 1 } , e _ { 2 })$ forms a basis of $\mathcal { E } ^ { * }$.
We equip $\mathcal { E } ^ { * }$ with the unique inner product making $(e _ { 1 } , e _ { 2 })$ an orthonormal basis.
b) Let $\mathcal { R } = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$. Show that the set $\mathcal { R }$ is a root system of $\mathcal { E } ^ { * }$. For this, you may draw the set $\mathcal { R }$ in the Euclidean space $\mathcal { E } ^ { * }$ and recognise one of the root systems encountered in question I.D.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Let $\alpha \in \mathcal { R }$. Determine by calculation the vector subspace $\mathcal { A } _ { \alpha }$. Verify that $\mathcal { A } _ { \alpha }$ is a one-dimensional vector space.

We use the notations from Parts I and II as well as from question III.B. We assume $$\mathcal { A } = \left\{ \left. \left( \begin{array} { c c } A & B \\ C & - { } ^ { t } A \end{array} \right) \right\rvert \, ( A , B , C ) \in ( \mathcal { M } ( 2 , \mathbb { R } ) ) ^ { 3 } , B = { } ^ { t } B \text { and } C = { } ^ { t } C \right\}$$ and $\mathcal { E } = \left\{ \left. \left( \begin{array} { c c } D & 0 \\ 0 & - D \end{array} \right) \right\rvert \, D \in \mathcal { D } ( 2 , \mathbb { R } ) \right\}$, with $\mathcal{R} = \left\{ e _ { 1 } - e _ { 2 } , e _ { 2 } - e _ { 1 } , e _ { 1 } + e _ { 2 } , - e _ { 1 } - e _ { 2 } , 2 e _ { 1 } , - 2 e _ { 1 } , 2 e _ { 2 } , - 2 e _ { 2 } \right\}$.
Establish the relation $\mathcal { A } = \mathcal { A } _ { 0 } \oplus \bigoplus _ { \alpha \in \mathcal { R } } \mathcal { A } _ { \alpha }$.

Let $W_{\ell} = \bigoplus_{0 \leq i < \ell} \mathbf{C} v_i$ and $a \in \mathbf{C}^*$. We consider the element $G_a$ of $\mathcal{L}(W_{\ell})$ whose matrix in the basis $\{v_i\}_{0 \leq i < \ell}$ is: $$\left(\begin{array}{cccccc} 0 & 0 & 0 & \cdots & 0 & a \\ 1 & 0 & 0 & \cdots & 0 & 0 \\ 0 & 1 & 0 & \cdots & 0 & 0 \\ \vdots & 0 & \ddots & \ddots & \vdots & \vdots \\ 0 & \vdots & \ddots & \ddots & 0 & 0 \\ 0 & 0 & \ldots & 0 & 1 & 0 \end{array}\right)$$
10a. Calculate $G_a^{\ell}$. Show that $G_a$ is diagonalizable.
10b. Let $b$ be an $\ell$-th root of $a$. Calculate the eigenvectors of $G_a$ and the associated eigenvalues in terms of $b, q$ and the $v_i$.

a) Show that the set of harmonic polynomials is a vector subspace of $\mathcal{P}$.
b) For all $m \geqslant 2$, we denote by $\Delta_m$ the restriction of $\Delta$ to $\mathcal{P}_m$. Show that $\operatorname{dim}(\operatorname{ker} \Delta_m) \geqslant 2m+1$.
c) What can be deduced about the dimension of the vector space of harmonic polynomials?

Determine a harmonic polynomial $H$ that satisfies $H(x,y) = f(x,y)$ for all $(x,y) \in C(0,1)$, where $f(x,y) = xy$.

Determine a harmonic polynomial $H$ that satisfies $H(x,y) = f(x,y)$ for all $(x,y) \in C(0,1)$, where $f(x,y) = x^4 - y^4$.

Let $x$ be a nonzero vector of $E$. Show that there exists a strictly positive integer $p$ such that the family $\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is free and that there exists $\left(\alpha_0, \alpha_1, \ldots, \alpha_{p-1}\right) \in \mathbb{K}^p$ such that:
$$\alpha_0 x + \alpha_1 f(x) + \cdots + \alpha_{p-1} f^{p-1}(x) + f^p(x) = 0.$$

Let $x$ be a non-zero vector of $E$. Show that there exists a strictly positive integer $p$ such that the family $\left(x, f(x), f^2(x), \ldots, f^{p-1}(x)\right)$ is free and that there exists $\left(\alpha_0, \alpha_1, \ldots, \alpha_{p-1}\right) \in \mathbb{K}^p$ such that:
$$\alpha_0 x + \alpha_1 f(x) + \cdots + \alpha_{p-1} f^{p-1}(x) + f^p(x) = 0.$$

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$, $\mathcal{Z} := \{u \in \mathcal{W} : \bar{u} = 0\}$, and $L$ the vector subspace such that $\mathcal{Z} = \{a \otimes x \mid a \in L\}$.
Prove that
$$\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}, \quad \operatorname{dim}(\operatorname{Vect}(x) \oplus \mathcal{V} x) + \operatorname{dim} L = n$$
and
$$L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$$
Deduce that $\operatorname{Vect}(x) \oplus \mathcal{V} x$ contains $v^{k}(x)$ for every $v \in \mathcal{V}$ and every $k \in \mathbf{N}$.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We consider an arbitrary vector $x$ of $E \backslash \{0\}$, and set $H := \operatorname{Vect}(x)^{\perp}$, $\mathcal{V} x := \{v(x) \mid v \in \mathcal{V}\}$, $\mathcal{W} := \{v \in \mathcal{V} : v(x) = 0\}$, $\overline{\mathcal{V}} := \{\bar{u} \mid u \in \mathcal{W}\}$. We have established that $\operatorname{dim} \overline{\mathcal{V}} = \frac{(n-1)(n-2)}{2}$ and $L^{\perp} = \operatorname{Vect}(x) \oplus \mathcal{V} x$.
By applying the induction hypothesis, show that the generic nilindex of $\mathcal{V}$ is greater than or equal to $n-1$, and that if moreover $\mathcal{V} x = \{0\}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.

We fix a real vector space $E$ of dimension $n \geq 2$, as well as a nilpotent vector subspace $\mathcal{V}$ of $\mathcal{L}(E)$ with $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$, equipped with an inner product $(-\mid-)$. We choose $x$ in $\mathcal{V}^{\bullet} \backslash \{0\}$. We denote by $p$ the generic nilindex of $\mathcal{V}$ (with $p \geq n-1$), and we fix $u \in \mathcal{V}$ such that $x \in \operatorname{Im} u^{p-1}$. We have established that $K(\mathcal{V}) = \operatorname{Vect}(\mathcal{V}^{\bullet}) \subset \operatorname{Vect}(x) \oplus \mathcal{V} x$ (from question 23 applied to all $v \in \mathcal{V}$), and Lemma B states that if $K(\mathcal{V}) \subset \operatorname{Vect}(x) + \mathcal{V} x$ then $v(x) = 0$ for every $v \in \mathcal{V}$.
Conclude the proof of Gerstenhaber's theorem: if $\operatorname{dim} \mathcal{V} = \frac{n(n-1)}{2}$ then there exists a basis of $E$ in which every element of $\mathcal{V}$ is represented by a strictly upper triangular matrix.

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 $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathscr{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha}).$$
(a) Show that for all $\omega \in \mathscr{A}_p(E, \mathbb{R})$, we have $\omega = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \Omega_p(e_{\alpha})$.
(b) Deduce that $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ is an inner product on $\mathscr{A}_p(E, \mathbb{R})$ for which $(\Omega_p(e_{\alpha}))_{\alpha \in \mathcal{I}_p}$ is an orthonormal basis of $\mathscr{A}_p(E, \mathbb{R})$ and give the dimension of $\mathscr{A}_p(E, \mathbb{R})$.
(c) Construct in the case $p = d-1$ an isometry between $\mathscr{A}_p(E, \mathbb{R})$ and $E$.

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 $\alpha = (i_1, \ldots, i_p) \in \mathcal{I}_p$, we denote $e_{\alpha} = (e_{i_1}, \ldots, e_{i_p}) \in E^p$ and for all $\omega$ and $\omega^{\prime}$ elements of $\mathcal{A}_p(E, \mathbb{R})$ $$\langle\omega, \omega^{\prime}\rangle = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \omega^{\prime}(e_{\alpha})$$
(a) Show that for all $\omega \in \mathcal{A}_p(E, \mathbb{R})$, we have $\omega = \sum_{\alpha \in \mathcal{I}_p} \omega(e_{\alpha}) \Omega_p(e_{\alpha})$.
(b) Deduce that $(\omega, \omega^{\prime}) \mapsto \langle\omega, \omega^{\prime}\rangle$ is an inner product on $\mathcal{A}_p(E, \mathbb{R})$ for which $(\Omega_p(e_{\alpha}))_{\alpha \in \mathcal{I}_p}$ is an orthonormal basis of $\mathcal{A}_p(E, \mathbb{R})$ and give the dimension of $\mathcal{A}_p(E, \mathbb{R})$.
(c) Construct in the case $p = d-1$ an isometry between $\mathcal{A}_p(E, \mathbb{R})$ and $E$.

Let $n$ be a non-zero natural number. We denote $E _ { n } = \mathbb { R } _ { n } [ X ]$ and for all $k \in \llbracket 0 , n \rrbracket , P _ { k } = X ^ { k }$.
Let $\alpha$ be a real number.

1. Justify that the family $\mathcal { E } = \left( 1 , X - \alpha , \ldots , ( X - \alpha ) ^ { n } \right)$ is a basis of $E _ { n }$.
2. Let $P$ be a polynomial in $E _ { n }$. Give without proof the decomposition of $P$ in the basis $\mathcal { E }$ using the successive derivatives of the polynomial $P$.
3. Suppose that $\alpha$ is a root of order $r \in \llbracket 1 , n \rrbracket$ of $P$. Determine the quotient and remainder of the Euclidean division of $P$ by $( X - \alpha ) ^ { r }$.

To every polynomial $P$ of $E _ { n }$, we associate the polynomial $Q$ defined by: $$Q ( X ) = X P ( X ) - \frac { 1 } { n } \left( X ^ { 2 } - 1 \right) P ^ { \prime } ( X )$$ and we denote by $T$ the application that associates $Q$ to $P$.

1. Let $k \in \llbracket 0 , n \rrbracket$. Determine $T \left( P _ { k } \right)$.
2. Show that $T$ is an endomorphism of $E _ { n }$.
3. Write the matrix $M$ of $T$ in the basis $\mathscr { B } = \left( P _ { 0 } , P _ { 1 } , \ldots , P _ { n } \right)$ of $E _ { n }$.
4. Suppose that $\lambda$ is a real eigenvalue of the endomorphism $T$ and let $P$ be a monic polynomial, eigenvector associated with the eigenvalue $\lambda$.
   1. [7.1.] Show that $P$ has degree $n$.
   2. [7.2.] Let $z _ { 0 }$ be a complex root of $P$ with multiplicity order $r \in \mathbb { N } ^ { * }$. Prove that $z _ { 0 } ^ { 2 } - 1 = 0$.
   3. [7.3.] Deduce an expression for $P$.
5. Determine the eigenvectors of the endomorphism $T$. Is the endomorphism $T$ diagonalisable?

Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$. Throughout the following, we make the following hypothesis: every element of $V$ is a matrix of rank at most $r$. Show that we can assume that $V$ contains the block matrix: $$A = \left( \begin{array} { c c } I _ { r } & 0 \\ 0 & 0 \end{array} \right)$$

Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$ such that every element of $V$ is a matrix of rank at most $r$, and we assume $V$ contains the block matrix $A = \left( \begin{array} { c c } I _ { r } & 0 \\ 0 & 0 \end{array} \right)$. a) Let $B$ be an element of $V$, which we write in the form of a block matrix: $$B = \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \\ B _ { 21 } & B _ { 22 } \end{array} \right)$$ where the four matrices $B _ { 11 } , B _ { 12 } , B _ { 21 } , B _ { 22 }$ are respectively in $M _ { r } ( \mathbf { C } )$, $M _ { r , m - r } ( \mathbf { C } ) , M _ { m - r , r } ( \mathbf { C } )$ and $M _ { m - r } ( \mathbf { C } )$. Show that $B _ { 22 } = 0$ and $B _ { 21 } B _ { 12 } = 0$ (one may consider the minors of size $r + 1$ of the matrix $t A + B$ for $t \in \mathbf { C }$). b) Let $B$ and $C$ be two matrices of $V$, which we write in block matrix form as above: $$B = \left( \begin{array} { c c } B _ { 11 } & B _ { 12 } \\ B _ { 21 } & 0 \end{array} \right) ; \quad C = \left( \begin{array} { c c } C _ { 11 } & C _ { 12 } \\ C _ { 21 } & 0 \end{array} \right)$$ Show that $B _ { 21 } C _ { 12 } + C _ { 21 } B _ { 12 } = 0$.

Let $r$ and $m$ be strictly positive integers with $r \leq m$. We consider a subspace $V$ of the $\mathbf { C }$-vector space $M _ { m } ( \mathbf { C } )$ such that every element of $V$ is a matrix of rank at most $r$, and we assume $V$ contains the block matrix $A = \left( \begin{array} { c c } I _ { r } & 0 \\ 0 & 0 \end{array} \right)$. By question V.2a), every element $B$ of $V$ has the block form $B = \left( \begin{array} { c c } B_{11} & B_{12} \\ B_{21} & 0 \end{array} \right)$. We denote by $W$ the intersection of $V$ with the subspace of $M _ { m } ( \mathbf { C } )$ consisting of block matrices of the form $$\left( \begin{array} { c c } 0 & 0 \\ B _ { 21 } & 0 \end{array} \right)$$ We define a linear application $\varphi$ from $M _ { m } ( \mathbf { C } )$ to $M _ { r , m } ( \mathbf { C } )$ by $$\varphi : \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \\ B _ { 21 } & B _ { 22 } \end{array} \right) \mapsto \left( \begin{array} { l l } B _ { 11 } & B _ { 12 } \end{array} \right)$$ (with the notations of V.2a)). a) We write any matrix $C$ of $M _ { r , m } ( \mathbf { C } )$ in the form of a block matrix $C = \left( \begin{array} { l l } C _ { 11 } & C _ { 12 } \end{array} \right)$ with $C _ { 11 } \in M _ { r } ( \mathbf { C } )$ and $C _ { 12 } \in M _ { r , m - r } ( \mathbf { C } )$. Let $\psi$ be the linear map from $W$ to $M _ { r , m } ( \mathbf { C } ) ^ { \vee }$ which sends $B = \left( \begin{array} { c c } 0 & 0 \\ B _ { 21 } & 0 \end{array} \right)$ to the linear form $C \mapsto \operatorname { Tr } \left( B _ { 21 } C _ { 12 } \right)$. Let $s = \operatorname { dim } W$. Using the map $\psi$, show that $\operatorname { dim } ( \varphi ( V ) ) \leq m r - s$. b) Deduce that $\operatorname { dim } V \leq m r$.

a) Let $r , m , n$ be strictly positive integers such that $r \leq n \leq m$. Show that if $E$ is a subspace of $M _ { m , n } ( \mathbf { C } )$ such that every element of $E$ is a matrix of rank at most $r$, then $\operatorname { dim } E \leq m r$. b) Give an example of a subspace $E$ of $M _ { m , n } ( \mathbf { C } )$ satisfying $\operatorname { dim } E = m r$ and such that every element of $E$ is a matrix of rank at most $r$.