In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$.
I.B.1) Show that there exist at least two subspaces of $E$ stable by $f$ and give an example of an endomorphism of $\mathbb{R}^2$ which admits only two stable subspaces.
I.B.2) Show that if $E$ is of finite dimension $n \geqslant 2$ and if $f$ is non-zero and non-injective, then there exist at least three subspaces of $E$ stable by $f$ and at least four when $n$ is odd.
Give an example of an endomorphism of $\mathbb{R}^2$ which admits only three stable subspaces.

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$.
I.C.1) Show that every subspace generated by a family of eigenvectors of $f$ is stable by $f$. Specify the endomorphism induced by $f$ on every eigenspace of $f$.
I.C.2) Show that if $f$ admits an eigenspace of dimension at least equal to 2 then there exist infinitely many lines of $E$ stable by $f$.
I.C.3) What can be said about $f$ if all subspaces of $E$ are stable by $f$?

In this problem, $\mathbb{K}$ denotes the field $\mathbb{R}$ or the field $\mathbb{C}$ and $E$ is a non-zero $\mathbb{K}$-vector space. $f$ is an endomorphism of a $\mathbb{K}$-vector space $E$. In this subsection, $E$ is a space of finite dimension.
I.D.1) Show that if $f$ is diagonalisable then every subspace of $E$ admits a complement in $E$ stable by $f$.
One may start from a basis of $F$ and a basis of $E$ consisting of eigenvectors of $f$.
I.D.2) Show that if $\mathbb{K} = \mathbb{C}$ and if every subspace of $E$ stable by $f$ admits a complement in $E$ stable by $f$, then $f$ is diagonalisable.
What about the case if $\mathbb{K} = \mathbb{R}$?

In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$.
The goal here is to show that a subspace $F$ of $E$ is stable by $f$ if and only if $F = \bigoplus_{i=1}^{p} (F \cap E_i)$.
II.A.1) Show that every subspace $F$ of $E$ such that $F = \bigoplus_{i=1}^{p} (F \cap E_i)$ is stable by $f$.
II.A.2) Let $F$ be a subspace of $E$ stable by $f$ and $x$ a non-zero vector of $F$.
Justify the existence and uniqueness of $(x_i)_{1 \leqslant i \leqslant p}$ in $E_1 \times \cdots \times E_p$ such that $x = \sum_{i=1}^{p} x_i$.
II.A.3) If we denote $H_x = \{i \in \llbracket 1, p \rrbracket \mid x_i \neq 0\}$, $H_x$ is non-empty and, up to reordering the eigenvalues (and the eigenspaces), we can assume that $H_x = \llbracket 1, r \rrbracket$ with $1 \leqslant r \leqslant p$. Thus we have $x = \sum_{i=1}^{r} x_i$ with $x_i \in E_i \setminus \{0\}$ for all $i$ in $\llbracket 1, r \rrbracket$.
We denote $V_x = \operatorname{Vect}(x_1, \ldots, x_r)$.
Show that $\mathcal{B}_x = (x_1, \ldots, x_r)$ is a basis of $V_x$.
II.A.4) Show that for all $j$ in $\llbracket 1, r \rrbracket$, $f^{j-1}(x)$ belongs to $V_x$ and give the matrix of the family $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ in the basis $\mathcal{B}_x$.
II.A.5) Show that $(f^{j-1}(x))_{1 \leqslant j \leqslant r}$ is a basis of $V_x$.
II.A.6) Deduce that for all $i$ in $\llbracket 1, p \rrbracket$, $x_i$ belongs to $F$ and conclude.

In this part, $n$ and $p$ are two natural integers at least equal to 2, $f$ is a diagonalisable endomorphism of a $\mathbb{K}$-vector space $E$ of dimension $n$, which admits $p$ distinct eigenvalues $\{\lambda_1, \ldots, \lambda_p\}$ and, for all $i$ in $\llbracket 1, p \rrbracket$, we denote by $E_i$ the eigenspace of $f$ associated with the eigenvalue $\lambda_i$. In this subsection, we place ourselves in the case where $p = n$.
II.B.1) Specify the dimension of $E_i$ for all $i$ in $\llbracket 1, p \rrbracket$.
II.B.2) How many lines of $E$ are stable by $f$?
II.B.3) If $n \geqslant 3$ and $k \in \llbracket 2, n-1 \rrbracket$, how many subspaces of $E$ of dimension $k$ and stable by $f$ are there?
II.B.4) How many subspaces of $E$ are stable by $f$ in this case? Give them all.

We consider the differentiation endomorphism $D$ on $\mathbb{K}[X]$ defined by $D(P) = P'$ for all $P$ in $\mathbb{K}[X]$.
III.A.1) Verify that for all $n$ in $\mathbb{N}$, $\mathbb{K}_n[X]$ is stable by $D$ and give the matrix $A_n$ of the endomorphism induced by $D$ on $\mathbb{K}_n[X]$ in the canonical basis of $\mathbb{K}_n[X]$.
III.A.2) Let $F$ be a subspace of $\mathbb{K}[X]$, of finite non-zero dimension, stable by $D$.
a) Justify the existence of a natural integer $n$ and a polynomial $R$ of degree $n$ such that $R \in F$ and $F \subset \mathbb{K}_n[X]$.
b) Show that the family $(D^i(R))_{0 \leqslant i \leqslant n}$ is a free family of $F$.
c) Deduce that $F = \mathbb{K}_n[X]$.
III.A.3) Give all subspaces of $\mathbb{K}[X]$ stable by $D$.

We consider an endomorphism $f$ of a $\mathbb{K}$-vector space $E$ of dimension $n \geqslant 2$ such that $f^n = 0$ and $f^{n-1} \neq 0$.
III.B.1) Determine the set of vectors $u$ of $E$ such that the family $\mathcal{B}_{f,u} = (f^{n-i}(u))_{1 \leqslant i \leqslant n}$ is a basis of $E$.
III.B.2) In the case where $\mathcal{B}_{f,u}$ is a basis of $E$, what is the matrix of $f$ in $\mathcal{B}_{f,u}$?
III.B.3) Determine a basis of $E$ such that the matrix of $f$ in this basis is $A_{n-1}$.
III.B.4) Give all subspaces of $E$ stable by $f$. How many are there? Give a simple relation between these stable subspaces and the kernels $\ker(f^i)$ for $i$ in $\llbracket 0, n \rrbracket$.

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
Show that if $n$ is odd, then $f$ admits at least one real eigenvalue.

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
In this question, $\lambda = \alpha + \mathrm{i}\beta$, with $(\alpha, \beta)$ in $\mathbb{R}^2$, is a non-real eigenvalue of $M$ and $Z$ in $\mathcal{M}_{n,1}(\mathbb{C})$, non-zero, is such that $MZ = \lambda Z$.
If $M = (m_{i,j})_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ we denote $\bar{M} = (m_{i,j}')_{\substack{1 \leqslant i \leqslant n \\ 1 \leqslant j \leqslant n}}$ with $m_{i,j}' = \bar{m}_{i,j}$ (conjugate of the complex number $m_{i,j}$) for all $(i,j)$ in $\llbracket 1, n \rrbracket^2$ and if $Z = \begin{pmatrix} z_1 \\ \vdots \\ z_n \end{pmatrix}$ we denote $\bar{Z} = \begin{pmatrix} z_1' \\ \vdots \\ z_n' \end{pmatrix}$ with $z_i' = \bar{z}_i$ for all $i$ in $\llbracket 1, n \rrbracket$.
We set $X = \frac{1}{2}(Z + \bar{Z})$ and $Y = \frac{1}{2\mathrm{i}}(Z - \bar{Z})$.
IV.C.1) Verify that $X$ and $Y$ are in $E$ and show that the family $(X, Y)$ is free in $E$.
IV.C.2) Show that the vector plane $F$ generated by $X$ and $Y$ is stable by $f$ and give the matrix of $f_F$ in the basis $(X, Y)$.

In this part, $n$ is a non-zero natural integer, $M$ is in $\mathcal{M}_n(\mathbb{R})$ and $f$ is the endomorphism of $E = \mathcal{M}_{n,1}(\mathbb{R})$ defined by $f(X) = MX$ for all $X$ in $E$.
What do you think of the statement: ``every endomorphism of a finite-dimensional real vector space admits at least one line or one plane stable''?

Does there exist an endomorphism of $\mathbb{R}[X]$ admitting neither line nor plane stable?

In this question, $E$ is a real vector space of dimension $n$ and $f$ is an endomorphism of $E$.
V.D.1) Show that if $f$ is diagonalisable then there exist $n$ hyperplanes of $E$, $(H_i)_{1 \leqslant i \leqslant n}$, all stable by $f$ such that $\bigcap_{i=1}^{n} H_i = \{0\}$.
V.D.2) Is an endomorphism $f$ of $E$ for which there exist $n$ hyperplanes of $E$ stable by $f$ and with intersection reduced to the zero vector necessarily diagonalisable?

Let $M$ and $N$ be in $S _ { n } ( \mathbb { R } )$. Show that there exists $U \in O _ { n } ( \mathbb { R } )$ such that $N = U M U ^ { - 1 }$, if and only if $\chi _ { M } = \chi _ { N }$.

Specify the set of eigenvalues of $\tau$. Is the application $\tau$ diagonalizable?

We consider the family of polynomials $$\left\{ \begin{array}{l} H_0 = 1 \\ H_k = \frac{1}{k!} \prod_{j=0}^{k-1} (X - j) \quad \text{for } k \in \llbracket 1, n \rrbracket \end{array} \right.$$
Is the matrix $M$ defined in question I.A.3 and the matrix $M'$ of size $n+1$ given by $$M' = \left(\begin{array}{ccccc} 1 & 1 & 0 & \ldots & 0 \\ 0 & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & 0 \\ \vdots & & \ddots & \ddots & 1 \\ 0 & \ldots & \ldots & 0 & 1 \end{array}\right)$$ similar?

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part.
Let $A \in \mathcal{M}_{2}(\mathbb{R})$. Show that $A$ is positively stable if and only if $\operatorname{tr}(A) > 0$ and $\operatorname{det}(A) > 0$.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part.
a) Is the sum of two positively stable matrices of $\mathcal{M}_{2}(\mathbb{R})$ necessarily positively stable?
b) Let $A, B$ in $\mathcal{M}_{n}(\mathbb{R})$ be two positively stable matrices that commute. Show that $A + B$ is positively stable.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ such that $A_{s}$ is positive definite.
a) Let $X = Y + \mathrm{i}Z$ be a column matrix of $\mathcal{M}_{n,1}(\mathbb{C})$, where $Y$ and $Z$ belong to $\mathcal{M}_{n,1}(\mathbb{R})$. We set $\bar{X} = Y - \mathrm{i}Z$ and we identify the matrix $\bar{X}^{\top}AX \in \mathcal{M}_{1}(\mathbb{C})$ with the complex number equal to its unique entry.
Show that, if $X \neq 0$, then $\operatorname{Re}(\bar{X}^{\top}AX) > 0$, where $\operatorname{Re}(z)$ denotes the real part of $z \in \mathbb{C}$.
b) Show that $A$ is positively stable.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part.
Give an example of a positively stable matrix $A$ such that $A_{s}$ is not positive definite.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. We consider the endomorphism $\Phi$ of $\mathcal{M}_{n}(\mathbb{R})$ such that $$\forall M \in \mathcal{M}_{n}(\mathbb{R}), \quad \Phi(M) = A^{\top}M + MA$$
Show that $\Phi$ is positively stable, that is, its matrix in any basis of $\mathcal{M}_{n}(\mathbb{R})$ is positively stable.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. We consider the endomorphism $\Phi$ of $\mathcal{M}_{n}(\mathbb{R})$ such that $\forall M \in \mathcal{M}_{n}(\mathbb{R}), \Phi(M) = A^{\top}M + MA$.
a) Show that there exists a unique matrix $B \in \mathcal{M}_{n}(\mathbb{R})$ such that $A^{\top}B + BA = I_{n}$.
b) Show that $B$ is symmetric and that $\operatorname{det}(B) > 0$.

A matrix $A$ of $\mathcal{M}_{n}(\mathbb{R})$ is said to be positively stable if all its complex eigenvalues have strictly positive real part. Let $A \in \mathcal{M}_{n}(\mathbb{R})$ be a positively stable matrix. Recall that $\exp(M) = \sum_{k=0}^{\infty} \frac{M^{k}}{k!}$ for any $M \in \mathcal{M}_{n}(\mathbb{C})$. For all real $t$, we set $V(t) = \exp(-tA^{\top})\exp(-tA)$ and $W(t) = \int_{0}^{t} V(s)\,\mathrm{d}s$.
a) Show that, for all real $t$, $V(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$ and that, if $t > 0$, $W(t) \in \mathcal{S}_{n}^{++}(\mathbb{R})$.
b) Show that, for all real $t$, $A^{\top}W(t) + W(t)A = I_{n} - V(t)$.
c) What do we obtain by letting $t$ tend to $+\infty$ in the previous equality? Deduce that the matrix $B$ of question III.C.2 is positive definite.

For every real eigenvalue $\lambda$ of $A$, show that $\min \operatorname{sp}_{\mathbb{R}}\left(A_{s}\right) \leqslant \lambda \leqslant \max \operatorname{sp}_{\mathbb{R}}\left(A_{s}\right)$.
Deduce that if $A_{s} \in \mathcal{S}_{n}^{++}(\mathbb{R})$ then $A$ is invertible.

Let $A \in \mathrm{O}_{n}(\mathbb{R})$. Show that the eigenvalues of $A_{s}$ are in $[-1,1]$.

Let $S \in \mathcal{S}_{n}(\mathbb{R})$.
a) We assume that $\operatorname{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and that for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension. Show that there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$.
b) Conversely, show that if there exists $A \in \mathrm{O}_{n}(\mathbb{R})$ such that $A_{s} = S$, then $\mathrm{sp}_{\mathbb{R}}(S) \subset [-1,1]$ and for every eigenvalue $\lambda$ of $S$ in $]-1,1[$, the eigenspace of $S$ associated with $\lambda$ has even dimension.