We keep the notations from the previous parts. In particular, we still denote by $x _ { k }$ the minimizer of $J$ on $x _ { 0 } + H _ { k }$. We set $H _ { 0 } = \{ 0 \}$ and for $k \geq 1$, $$H _ { k } = \left\{ P ( A ) r _ { 0 } \mid P \in \mathbb { R } [ X ] , \operatorname { deg } ( P ) \leq k - 1 \right\}$$
Show that there exists a family $\left( p _ { 0 } , \ldots , p _ { m - 1 } \right)$ of vectors in $\mathbb { R } ^ { N }$ such that
(i) For all $k \in \{ 1 , \ldots , m \}$, the family $( p _ { 0 } , \ldots , p _ { k - 1 } )$ is a basis of $H _ { k }$.
(ii) The family is orthogonal with respect to the inner product associated with $A$, that is $$\forall i , j \in \{ 0 , \ldots , m - 1 \} \quad i \neq j \Rightarrow \left\langle A p _ { i } , p _ { j } \right\rangle = 0$$

We denote by $d$ the degree of $\pi_f$. Justify the existence of a vector $x_1$ of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free.

We denote by $d$ the degree of $\pi_f$. Justify the existence of a vector $x_1$ of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free.

Let $V \in \mathcal{M}_n(\mathbb{C})$ be a nilpotent matrix of index $p$. We propose to study the equation $R^2 = V$.
Show that, if $2p - 1 > n$, then there is no solution.

Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$.
Let $M \in \mathcal{M}_{n}(\mathbb{R})$. Show that $M$ belongs to $\mathcal{A}$ if and only if, for all $i \in \llbracket 1, r \rrbracket$, $\langle A_{i} \mid M \rangle = 0$.

We keep the notations from the previous parts. In particular, we still denote by $x _ { k }$ the minimizer of $J$ on $x _ { 0 } + H _ { k }$.
Assume that a family $\left( p _ { 0 } , \ldots , p _ { m - 1 } \right)$ of vectors satisfying the properties of question 24 is known. Show that $x _ { k + 1 } - x _ { k }$ is then collinear with $p _ { k }$ for all integer $k \in \{ 0 , \ldots , m - 1 \}$.

We set $e_1 = x_1, e_2 = f(x_1), \ldots, e_d = f^{d-1}(x_1)$ and $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$. Show that $E_1$ is stable under $f$ and that $E_1 = \{P(f)(x_1) \mid P \in \mathbb{K}[X]\}$.

We denote by $d$ the degree of $\pi_f$, and $x_1$ is a vector of $E$ such that $\left(x_1, f(x_1), \ldots, f^{d-1}(x_1)\right)$ is free. We set $e_1 = x_1, e_2 = f(x_1), \ldots, e_d = f^{d-1}(x_1)$ and $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$.
Show that $E_1$ is stable under $f$ and that $E_1 = \{P(f)(x_1) \mid P \in \mathbb{K}[X]\}$.

Let $V \in \mathcal{M}_n(\mathbb{C})$ be a nilpotent matrix of index $p$. We propose to study the equation $R^2 = V$.
For every value of the integer $n \geqslant 3$, exhibit a matrix $V \in \mathcal{M}_n(\mathbb{C})$, nilpotent of index $p \geqslant 2$ and admitting at least one square root.

Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$.
Show that for every matrix $N \in \mathcal{A}$ and all $i \in \llbracket 1, r \rrbracket$, we have $N^{\top} A_{i} \in \mathcal{A}^{\perp}$.

We set $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$ where $e_i = f^{i-1}(x_1)$, and we denote by $\psi_1$ the endomorphism induced by $f$ on the vector subspace $E_1$,
$$\psi_1 : \left\lvert\, \begin{aligned} & E_1 \rightarrow E_1, \\ & x \mapsto f(x). \end{aligned} \right.$$
Justify that $\psi_1$ is cyclic.

We denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(x_1, f(x_1), \ldots, f^{d-1}(x_1))$, and $\psi_1$ is the endomorphism induced by $f$ on the vector subspace $E_1$,
$$\psi_1 : \left\lvert\, \begin{aligned} & E_1 \rightarrow E_1, \\ & x \mapsto f(x). \end{aligned} \right.$$
Justify that $\psi_1$ is cyclic.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Prove that $\operatorname{Im} u$ is stable under $u$ and that the endomorphism induced by $u$ on $\operatorname{Im} u$ is nilpotent. Specify its nilpotency index.

Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$.
Show that $\mathcal{A}^{\top}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ of the same dimension as $\mathcal{A}$.

We complete, if necessary, $(e_1, e_2, \ldots, e_d)$ to a basis $(e_1, e_2, \ldots, e_n)$ of $E$. Let $\Phi$ be the $d$-th coordinate form which associates to any vector $x$ of $E$ its coordinate along $e_d$. We denote by $F = \{x \in E \mid \forall i \in \mathbb{N}, \Phi(f^i(x)) = 0\}$.
Show that $F$ is stable under $f$ and that $E_1$ and $F$ are in direct sum.

We denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$ where $e_i = f^{i-1}(x_1)$. We complete $(e_1, e_2, \ldots, e_d)$ to a basis $(e_1, e_2, \ldots, e_n)$ of $E$. Let $\Phi$ be the $d$-th coordinate form which associates to any vector $x$ of $E$ its coordinate along $e_d$. We denote by $F = \{x \in E \mid \forall i \in \mathbb{N}, \Phi(f^i(x)) = 0\}$.
Show that $F$ is stable under $f$ and that $E_1$ and $F$ are in direct sum.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
For every non-zero vector $x$ in $E$, we denote by $C_u(x)$ the vector space spanned by the $\left(u^k(x)\right)_{k \in \mathbb{N}}$; prove that $C_u(x)$ is stable under $u$ and that there exists a smallest integer $s(x) \geqslant 1$ such that $u^{s(x)}(x) = 0$.

Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$. Let $\mathcal{A}^{\top} = \{ M^{\top} \mid M \in \mathcal{A} \}$. We denote by $\mathcal{M}_{n,1}(\mathbb{R})$ the $\mathbb{R}$-vector space of column matrices with $n$ rows and real coefficients.
Let $X \in \mathcal{M}_{n,1}(\mathbb{R})$ and let $F = \operatorname{Vect}(A_{1}X, \ldots, A_{r}X)$. Show that $F$ is stable by the endomorphisms of $\mathcal{M}_{n,1}(\mathbb{R})$ canonically associated with elements of $\mathcal{A}^{\top}$.

We complete, if necessary, $(e_1, e_2, \ldots, e_d)$ to a basis $(e_1, e_2, \ldots, e_n)$ of $E$. Let $\Phi$ be the $d$-th coordinate form which associates to any vector $x$ of $E$ its coordinate along $e_d$. Let $\Psi$ be the linear map from $E$ to $\mathbb{K}^d$ defined, for all $x \in E$, by
$$\Psi(x) = \left(\Phi\left(f^i(x)\right)\right)_{0 \leqslant i \leqslant d-1} = \left(\Phi(x), \Phi(f(x)) \ldots, \Phi\left(f^{d-1}(x)\right)\right)$$
Show that $\Psi$ induces an isomorphism between $E_1$ and $\mathbb{K}^d$.

We denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$ where $e_i = f^{i-1}(x_1)$, and $\Phi$ is the $d$-th coordinate form. Let $\Psi$ be the linear map from $E$ to $\mathbb{K}^d$ defined, for all $x \in E$, by
$$\Psi(x) = \left(\Phi\left(f^i(x)\right)\right)_{0 \leqslant i \leqslant d-1} = \left(\Phi(x), \Phi(f(x)) \ldots, \Phi\left(f^{d-1}(x)\right)\right)$$
Show that $\Psi$ induces an isomorphism between $E_1$ and $\mathbb{K}^d$.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$. For every non-zero vector $x$ in $E$, we denote by $C_u(x)$ the vector space spanned by the $\left(u^k(x)\right)_{k \in \mathbb{N}}$ and $s(x)$ the smallest integer $\geqslant 1$ such that $u^{s(x)}(x) = 0$.
Prove that $(x, u(x), \ldots, u^{s(x)-1}(x))$ is a basis of $C_u(x)$ and give the matrix, in this basis, of the endomorphism induced by $u$ on $C_u(x)$.

Throughout this part, $\mathcal{A}$ is a subalgebra of $\mathcal{M}_{n}(\mathbb{R})$ strictly contained in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $d$ its dimension. We denote by $\mathcal{A}^{\perp}$ the orthogonal complement of $\mathcal{A}$ in $\mathcal{M}_{n}(\mathbb{R})$ and we denote by $r$ its dimension. We fix a basis $(A_{1}, \ldots, A_{r})$ of $\mathcal{A}^{\perp}$.
Show that $d \leqslant n^{2} - n + 1$ and conclude.

Using the notation of Q28 and Q29, show that $E = E_1 \oplus F$.

We denote by $d$ the degree of $\pi_f$, $E_1 = \operatorname{Vect}(e_1, e_2, \ldots, e_d)$ where $e_i = f^{i-1}(x_1)$, $F = \{x \in E \mid \forall i \in \mathbb{N}, \Phi(f^i(x)) = 0\}$, and $\Psi$ is the linear map from $E$ to $\mathbb{K}^d$ defined by $\Psi(x) = \left(\Phi(f^i(x))\right)_{0 \leqslant i \leqslant d-1}$.
Show that $E = E_1 \oplus F$.

We assume $n \geqslant 2$. Let $u$ be an endomorphism of $E$ nilpotent of index $p \geqslant 2$.
Prove by induction on $p$ that there exist vectors $x_1, \ldots, x_t$ in $E$ such that $E = \bigoplus_{i=1}^{t} C_u\left(x_i\right)$. One may apply the induction hypothesis to the endomorphism induced by $u$ on $\operatorname{Im}(u)$.