We consider two matrices of $M _ { 2 } ( \mathbf { Z } )$ that are $\mathbf { C }$-equivalent. Are they always $\mathbf { Z }$-equivalent?

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 $r$.

Let $a \in \mathbb{K}$. For all $p \in \mathbb{K}[X]$, we set $E_a(p) = E_a p = p(X+a)$.
Show that $E_a$ is an automorphism of $\mathbb{K}[X]$.

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?

After justifying the existence of the suprema, show that: $$\sup _ { \substack { x \in E \\ x \neq 0 } } \frac { \| u ( x ) \| } { \| x \| } = \sup _ { \substack { x \in E \\ \| x \| = 1 } } \| u ( x ) \| .$$

Show that a matrix $S \in S_n(\mathrm{R})$ belongs to $S_n^+(\mathrm{R})$ if, and only if, $\mathrm{Sp}(S) \subset \mathbf{R}_+$.
Similarly, we will admit in the rest of the problem that: $S \in S_n^{++}(\mathrm{R})$ if, and only if, $\operatorname{Sp}(S) \subset \mathbf{R}_+^\star$.

Show that a matrix $S \in S _ { n } ( \mathbf { R } )$ belongs to $S _ { n } ^ { + } ( \mathbf { R } )$ if, and only if, $\operatorname { Sp } ( S ) \subset \mathbf { R } _ { + }$.

Let $A \in \mathscr{M}_{N}(\mathbf{R})$. Show that $A$ satisfies $(M_2)$ if and only if $AU = U$. Deduce that if $A$ and $B$ are two Markov kernels then $AB$ is also a Markov kernel.

If $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, the support of $\varphi$ is defined by: $$\operatorname{Supp}(\varphi) = \overline{\{x \in \mathbb{R} : \varphi(x) \neq 0\}}$$ We say that $\varphi$ has compact support if $\operatorname{Supp}(\varphi)$ is a bounded subset of $\mathbb{R}$. We denote by $\mathcal{C}_{c}(\mathbb{R})$ the set of continuous functions with compact support on $\mathbb{R}$.
Show that $\mathcal{C}_{c}(\mathbb{R})$ is a vector subspace of the space of continuous functions on $\mathbb{R}$.

Recall the cardinality of $\mathcal{S}_n$. Deduce that $R \geq 1$.

a) Show that $\mathbb{H}$ is a sub-$\mathbb{R}$-algebra of $M_2(\mathbb{C})$ stable under $Z \mapsto Z^*$. b) Let $Z \in \mathbb{H}$. Calculate $ZZ^*$ and deduce that every non-zero element of $\mathbb{H}$ is invertible. c) Let $Z \in \mathbb{H}$. Show that $Z \in \mathbb{R}_{\mathbb{H}}$ if and only if $ZZ' = Z'Z$ for all $Z' \in \mathbb{H}$.

Let $C \subset E$ be a convex set. Let $f$ and $g$ be two convex functions from $C$ to $\mathbb{R}$.
(a) Show that $f + g$ is convex, and strictly convex if one of the two functions $f$ or $g$ is strictly convex.
(b) Assume $f$ is strictly convex. Verify that the minimum of $f$ is attained on $C$ at most at one point of $C$.

a) Show that $N(ZZ') = N(Z)N(Z')$ for all $Z, Z' \in \mathbb{H}$. b) Show that $S$ is a subgroup of $\mathbb{H}^\times$ and that $\frac{1}{\sqrt{N(Z)}}Z \in S$ for all $Z \in \mathbb{H}^\times$.

Show that $\alpha$ is a group morphism and describe its kernel, where $$\begin{aligned} \alpha : S \times S & \longrightarrow \mathrm{GL}(\mathbb{H}) \\ (u, v) & \longmapsto (Z \mapsto uZv^{-1}) \end{aligned}$$

Show that the set of shift-invariant endomorphisms of $\mathbb{K}[X]$ is a subalgebra of $\mathcal{L}(\mathbb{K}[X])$. Is the set of delta endomorphisms of $\mathbb{K}[X]$ closed under addition? under composition?

Show that $\alpha$ is continuous and that the image of $\alpha$ is contained in $\mathrm{SO}(\mathbb{H})$. One may begin by showing that $\alpha(u,v) \in \mathrm{O}(\mathbb{H})$ for $(u,v) \in S \times S$.

Let $\theta \in \mathbb{R}$ and $v \in \mathbb{H}^{\mathrm{im}} \cap S$, and let $u = (\cos\theta)E + (\sin\theta)v$. a) Show that $u \in S$ and that $u^{-1} = (\cos\theta)E - (\sin\theta)v$. b) Let $w \in \mathbb{H}^{\mathrm{im}} \cap S$ be a vector orthogonal to $v$. Describe the matrix of $C_u$ in the direct orthonormal basis $(v, w, vw)$ of $\mathbb{H}^{\mathrm{im}}$.

Show that the map $u \mapsto C_u$ induces a surjective group morphism $S \rightarrow \mathrm{SO}(\mathbb{H}^{\mathrm{im}})$ and describe its kernel.

a) Deduce that $\alpha(S \times S) = \mathrm{SO}(\mathbb{H})$. b) Show that $N := \alpha(S \times \{E\})$ is a subgroup of $\mathrm{SO}(\mathbb{H})$, then that $gng^{-1} \in N$ for all $n \in N$ and $g \in \mathrm{SO}(\mathbb{H})$ and that $\{\pm\mathrm{id}\} \subsetneq N \subsetneq \mathrm{SO}(\mathbb{H})$.

Show that $\mathrm{Aut}(\mathbb{H})$ is a subgroup of $\mathrm{GL}(\mathbb{H})$, containing $\alpha(u,u)$ for all $u \in S$.

Show that $(f(I), f(J), f(K))$ is a direct orthonormal basis of $\mathbb{H}^{\mathrm{im}}$ for all $f \in \mathrm{Aut}(\mathbb{H})$.

a) Show that the restriction map to $\mathbb{H}^{\mathrm{im}}$ induces a group isomorphism $$\mathrm{Aut}(\mathbb{H}) \simeq \mathrm{SO}(\mathbb{H}^{\mathrm{im}}).$$ b) Show that $$\mathrm{Aut}(\mathbb{H}) = \{\alpha(u,u) \mid u \in S\}.$$