Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We denote by $0 < \lambda _ { 1 } < \cdots < \lambda _ { d }$ the $d$ eigenvalues of $A$ (distinct pairwise) and $F _ { 1 } , \ldots , F _ { d }$ the associated eigenspaces. We consider the linear map from $\mathbb { R } ^ { N }$ to $\mathbb { R } ^ { N }$: $$x \mapsto \sum _ { i = 1 } ^ { d } \lambda _ { i } ^ { 1 / 2 } p _ { F _ { i } } ( x )$$ where $p _ { F _ { i } }$ is the orthogonal projection (for the canonical inner product) onto $F _ { i }$. We denote by $A ^ { 1 / 2 }$ the matrix associated with this linear map in the canonical basis.
a) We write $A = U D U ^ { T }$, where $D \in \mathcal { M } _ { N } ( \mathbb { R } )$ is the diagonal matrix containing the eigenvalues of $A$ in increasing order, with their multiplicities, and $U$ an orthogonal matrix. We denote by $D ^ { 1 / 2 }$ the diagonal matrix whose diagonal coefficients are the square roots of those of $D$. Show that $A ^ { 1 / 2 } = U D ^ { 1 / 2 } U ^ { T }$.
b) Show that $A ^ { 1 / 2 } \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, that $A ^ { 1 / 2 } A ^ { 1 / 2 } = A$, and that $A ^ { 1 / 2 }$ commutes with $A$.
c) Show that, for all $x \in \mathbb { R } ^ { N } , \| x \| _ { A } = \left\| A ^ { 1 / 2 } x \right\|$, where $\| x \| _ { A }$ is the norm defined in question 4.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Show that $\operatorname{Im} u \subset \operatorname{Ker} u$ and that $2r \leqslant n$.

We fix a polynomial $f \in \mathbb{C}[X]$ of degree $n \geq 1$. We consider a complex number $z \in \overline{\mathbb{D}}$ and we define the matrices $M \in \mathcal{M}_{n+1}(\mathbb{C})$ and $P \in \mathcal{M}_{n+1,1}(\mathbb{C})$ by $$M = \left(\begin{array}{cccccc} z & 0 & 0 & \ldots & 0 & \sqrt{1-|z|^2} \\ \sqrt{1-|z|^2} & 0 & 0 & \ldots & 0 & -\bar{z} \\ 0 & & & & & 0 \\ 0 & & & & & 0 \\ \vdots & & I_{n-1} & & & \vdots \\ 0 & & & & & 0 \end{array}\right)$$ and $$P = \left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)$$ Show that $|f(z)| \leq \|f(M)\|$.

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\Gamma(\mathbb{K})$ is a subalgebra of $\mathcal{M}_{2}(\mathbb{K})$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ where $\operatorname { deg } ( P ) \in \mathbb { N }$ denotes the degree of the polynomial $P$.
Show that the $H _ { k }$ form a sequence of vector subspaces of $\mathbb { R } ^ { N }$, and show that $H _ { k } \subset H _ { k + 1 }$ for all $k \in \mathbb { N }$.
a) Show that there necessarily exists $k$ such that $H _ { k + 1 } = H _ { k }$. We then denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$.
b) Show that $\operatorname { dim } \left( H _ { k } \right) = m$ for all $k \geq m$, and that $\operatorname { dim } \left( H _ { k } \right) = k$ for $k \leq m$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Assume that $\operatorname{Im} u = \operatorname{Ker} u$. Show that there exist vectors $e_1, e_2, \ldots, e_r$ in $E$ such that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right)\right)$ is a basis of $E$.

We fix a polynomial $f \in \mathbb{C}[X]$ of degree $n \geq 1$. We consider a complex number $z \in \overline{\mathbb{D}}$ and we define the matrices $M \in \mathcal{M}_{n+1}(\mathbb{C})$ and $P \in \mathcal{M}_{n+1,1}(\mathbb{C})$ by $$M = \left(\begin{array}{cccccc} z & 0 & 0 & \ldots & 0 & \sqrt{1-|z|^2} \\ \sqrt{1-|z|^2} & 0 & 0 & \ldots & 0 & -\bar{z} \\ 0 & & & & & 0 \\ 0 & & & & & 0 \\ \vdots & & I_{n-1} & & & \vdots \\ 0 & & & & & 0 \end{array}\right)$$ and $$P = \left(\begin{array}{c} 1 \\ 0 \\ \vdots \\ 0 \end{array}\right)$$ Prove Theorem 1: Let $f \in \mathbb{C}[X]$ be a polynomial. Then $$\sup_{z \in \overline{\mathbb{D}}} |f(z)| = \sup_{z \in \mathbb{S}} |f(z)|$$

Let $\Gamma(\mathbb{K})$ be the subset of $\mathcal{M}_{2}(\mathbb{K})$ consisting of matrices of the form $\left( \begin{array}{cc} a & -b \\ b & a \end{array} \right)$ where $(a, b) \in \mathbb{K}^{2}$.
Show that $\Gamma(\mathbb{R})$ is not a diagonalisable subalgebra of $\mathcal{M}_{2}(\mathbb{R})$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$. We denote by $d$ the number of distinct eigenvalues of $A$.
a) In the special case where $r _ { 0 }$ is an eigenvector of $A$, show that the integer $m$ is equal to 1.
b) In the general case, show that $m$ is less than or equal to $d$.
c) For any integer $n$ between 1 and $d$, construct an $x _ { 0 }$ such that the integer $m$ is equal to $n$.
d) Show that the set of $x _ { 0 }$ for which the dimension $m$ is exactly equal to $d$ is the complement of a finite union of sets of the form $\tilde { x } + E$, where $E$ is a vector space of dimension less than or equal to $N - 1$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume that $\operatorname{Im} u = \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right)\right)$ is a basis of $E$.
Give the matrix of $u$ in this basis.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$.
Show that there exists a polynomial $Q$ of degree $m$ such that $Q ( A ) e _ { 0 } = 0$, where $e _ { 0 } = x _ { 0 } - \tilde { x }$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$.
Assume $\operatorname{Im} u \neq \operatorname{Ker} u$. Show that there exist vectors $e_1, e_2, \ldots, e_r$ in $E$ and vectors $v_1, v_2, \ldots, v_{n-2r}$ belonging to $\operatorname{Ker} u$ such that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Specify the matrices $J$ and $J^{2}$. (One may distinguish the cases $n = 2$ and $n > 2$.)

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$, and $Q$ is the polynomial of degree $m$ from question 9 such that $Q ( A ) e _ { 0 } = 0$ where $e _ { 0 } = x _ { 0 } - \tilde { x }$.
Show that the polynomial $Q$ satisfies $Q ( 0 ) \neq 0$.

We assume that $n \geqslant 3$. Let $u$ be an endomorphism of $E$ nilpotent of index 2 and of rank $r$. Assume $\operatorname{Im} u \neq \operatorname{Ker} u$ and that $\left(e_1, u\left(e_1\right), e_2, u\left(e_2\right), \ldots, e_r, u\left(e_r\right), v_1, \ldots, v_{n-2r}\right)$ is a basis of $E$.
What is the matrix of $u$ in this basis?

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
Specify the matrices $J^{n}$ and $J^{k}$ for $2 \leqslant k \leqslant n-1$.

Let $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$. We are given $b \in \mathbb { R } ^ { N }$ and we denote by $\tilde { x } \in \mathbb { R } ^ { N }$ the unique vector satisfying $A \tilde { x } = b$. We are given a vector $x _ { 0 } \in \mathbb { R } ^ { N }$, different from $\tilde { x }$, and we denote by $r _ { 0 } = b - A x _ { 0 }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$ We denote by $m$ the smallest integer $k$ such that $H _ { k + 1 } = H _ { k }$. We define $x _ { 0 } + H _ { k }$ as the subset of points of $\mathbb { R } ^ { N }$ of the form $x _ { 0 } + x$ where $x$ ranges over the vector space $H _ { k }$.
a) Show that $\tilde { x } \in x _ { 0 } + H _ { m }$.
b) Show that, for all $k \in \{ 0 , \ldots , m - 1 \}$, we have $\tilde { x } \notin x _ { 0 } + H _ { k }$.

We assume that $f$ is a nilpotent endomorphism of $E$. We denote by $r$ the smallest natural integer such that $f^r = 0$. Show that $f$ is cyclic if and only if $r = n$. Specify the companion matrix.

We assume that $f$ is a nilpotent endomorphism of $E$. We denote by $r$ the smallest natural number such that $f^r = 0$. Show that $f$ is cyclic if and only if $r = n$. Specify the companion matrix.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
Show that, if $A$ is nilpotent, then 0 is the unique eigenvalue of $A$.

In this part, we assume $n \geqslant 2$. For all $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$, we set $$J(a_{0}, \ldots, a_{n-1}) = \left( \begin{array}{cccc} a_{0} & a_{n-1} & \cdots & a_{1} \\ a_{1} & a_{0} & \cdots & a_{2} \\ \vdots & \vdots & & \vdots \\ a_{n-1} & a_{n-2} & \cdots & a_{0} \end{array} \right)$$ The coefficient with index $(i,j)$ of $J(a_{0}, \ldots, a_{n-1})$ is $a_{i-j}$ if $i \geqslant j$ and $a_{i-j+n}$ if $i < j$. Let $\mathcal{A}$ be the set of matrices of $\mathcal{M}_{n}(\mathbb{R})$ of the form $J(a_{0}, \ldots, a_{n-1})$ where $(a_{0}, \ldots, a_{n-1}) \in \mathbb{R}^{n}$. Let $J \in \mathcal{M}_{n}(\mathbb{R})$ be the matrix canonically associated with the endomorphism $\varphi \in \mathcal{L}(\mathbb{R}^{n})$ defined by $\varphi: e_{j} \mapsto e_{j+1}$ if $j \in \{1, \ldots, n-1\}$ and $\varphi(e_{n}) = e_{1}$, where $(e_{1}, \ldots, e_{n})$ is the canonical basis of $\mathbb{R}^{n}$.
What is the relationship between the matrix $J(a_{0}, \ldots, a_{n-1})$ and the $J^{k}$, where $0 \leqslant k \leqslant n-1$?

We introduce the map $$\begin{aligned} J : \mathbb { R } ^ { N } & \rightarrow \mathbb { R } \\ x & \mapsto \frac { 1 } { 2 } \langle x , A x \rangle - \langle b , x \rangle \end{aligned}$$ where $A \in \mathcal { S } _ { N } ^ { + } ( \mathbb { R } )$, $b \in \mathbb { R } ^ { N }$, and $\tilde { x }$ is the unique vector satisfying $A \tilde { x } = b$.
For all $x \in \mathbb { R } ^ { N }$, express $\| x - \tilde { x } \| _ { A } ^ { 2 } = \langle x - \tilde { x } , A ( x - \tilde { x } ) \rangle$ in terms of $J ( \tilde { x } )$ and $J ( x )$ and deduce that $\tilde { x }$ is the unique minimizer of $J$ on $\mathbb { R } ^ { N }$, that is, $J ( \tilde { x } ) \leq J ( x )$ for all $x \in \mathbb { R } ^ { N }$, and that $\tilde { x }$ is the only point satisfying this property.

We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, we set $F_k = \ker\left(\left(f - \lambda_k \operatorname{Id}_E\right)^{m_k}\right)$.
Show that the vector subspaces $F_k$ are stable under $f$ and that $E = F_1 \oplus \cdots \oplus F_p$.

We assume that $\mathbb{K} = \mathbb{C}$, that $(\mathrm{Id}, f, f^2, \ldots, f^{n-1})$ is free, and we factor the characteristic polynomial of $f$ in the form
$$\chi_f(X) = \prod_{k=1}^{p} \left(X - \lambda_k\right)^{m_k}$$
where the $\lambda_k$ are the $p$ eigenvalues pairwise distinct of $f$ and the $m_k \in \mathbb{N}^*$ their respective multiplicities. For $k \in \llbracket 1, p \rrbracket$, we set $F_k = \ker\left(\left(f - \lambda_k \operatorname{Id}_E\right)^{m_k}\right)$.
Show that the vector subspaces $F_k$ are stable under $f$ and that $E = F_1 \oplus \cdots \oplus F_p$.

Let $A$ denote a matrix in $\mathcal{M}_n(\mathbb{C})$.
What are the matrices in $\mathcal{M}_n(\mathbb{C})$ that are both nilpotent and diagonalizable?