In this section, we consider in $\mathbb{R}^2$ a circle $\mathcal{C}(\Omega, r)$ with center $\Omega$ and radius $r$ and $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ a matrix with proper eigenvalue circle equal to $\mathcal{C}(\Omega, r)$.
In this question $\Omega = (0, \alpha)$ with $\alpha > 0$ and $r = \alpha/2$.
Specify the eigenvalues of $A$ and give a matrix $B$ whose off-diagonal entries are opposite and which is directly orthogonally similar to $A$, as well as an orthogonal decomposition of the endomorphism canonically associated with $B$. Make a drawing in the case where $\alpha = 6$ illustrating questions IV.C.2 and IV.C.3.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y,z)$ in $\mathbb{R}^3$, we denote by $\psi_A(x,y,z)$ the real part of the determinant of the matrix $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, where $\mathrm{i}$ is the complex affix of the point $J = (0,1)$.
Calculate $\psi_A(x,y,z)$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, and let $Z_A$ be the intersection of $\mathcal{H}_A$ with the plane $x = (a+d)/2$.
If the matrix $A$ has two non-real eigenvalues, how can one see the eigenvalues of $A$ on $\mathcal{H}_A$? (One may consider the intersection of $Z_A$ with the plane with equation $y = 0$.) Can one see a basis of eigenvectors using $\mathcal{H}_A$?

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
In the case where $A = \left(\begin{array}{rr} 1 & 7 \\ -1 & 3 \end{array}\right)$ make a perspective drawing illustrating what precedes.

We assume $\alpha = 1$. We denote $\|\cdot\|$ the norm associated with $S_1$, $T_k$ the unique polynomial eigenvector of $\varphi_1$ of degree $k$, of norm 1 and with positive leading coefficient, and $V_n(z) = U_{n+1}(z,-1)$. Deduce that, for all $n \in \mathbb{N}$, $V_n$ and $T_n$ are proportional. Explicitly state the proportionality coefficient.

Let $s$ be an endomorphism of $E$ such that $s \circ s = \operatorname{Id}_E$. We set $F = \operatorname{Ker}(s - \operatorname{Id}_E)$ and $G = \operatorname{Ker}(s + \operatorname{Id}_E)$. a) Show that $F$ and $G$ are two supplementary subspaces of $E$. b) Deduce that $s$ is a symmetry and specify its elements.

Let $s$ and $t$ be two symmetries of $E$ that anticommute, that is, such that $s \circ t + t \circ s = 0$. a) Prove the equalities $t(F_s) = G_s$ and $t(G_s) = F_s$. b) Deduce that $F_s$ and $G_s$ have the same dimension and that $n$ is even.

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that $$\left\{ \begin{aligned} \forall i & A_i^2 & = I_n \\ \forall i \neq j & A_i A_j + A_j A_i & = 0 \end{aligned} \right.$$ In both cases, $p$ is called the length of the H-system. Show that the length $p$ of an H-system of endomorphisms of $E$ is bounded above by $n^2$.

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ Similarly, we call an H-system of matrices of size $n$ any finite family $(A_1, \ldots, A_p)$ of matrices of $\mathcal{M}_n(\mathbb{C})$ such that $$\left\{ \begin{aligned} \forall i & A_i^2 & = I_n \\ \forall i \neq j & A_i A_j + A_j A_i & = 0 \end{aligned} \right.$$ In both cases, $p$ is called the length of the H-system. Show that the existence of an H-system $(S_1, \ldots, S_p)$ of $E$ is equivalent to the existence of an H-system of matrices of size $n$. Deduce that the length of an H-system of $E$ depends only on the dimension $n$ of $E$ and not on the space $E$.

We call an H-system of endomorphisms of $E$ any finite family of symmetries of $E$ that anticommute pairwise, that is, any finite family $(S_1, \ldots, S_p)$ of endomorphisms of $E$ such that $$\left\{ \begin{array}{lrl} \forall i & S_i \circ S_i & = \operatorname{Id}_E \\ \forall i \neq j & S_i \circ S_j + S_j \circ S_i & = 0 \end{array} \right.$$ We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Let $n$ be an odd integer. Prove that $p(n) = 1$.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. We assume here that $n$ is even and we set $n = 2m$. We consider:

• an H-system $(S_1, \ldots, S_p, T, U)$ of $E$,
• the subspace $E_0 = F_T = \operatorname{Ker}(T - \mathrm{Id})$,
• for $j \in \llbracket 1, p \rrbracket$, the endomorphism $R_j = \mathrm{i} U \circ S_j$ of $E$.

a) Show that, for all $j \in \llbracket 1, p \rrbracket$, the subspace $E_0$ is stable under $R_j$. b) For $j \in \llbracket 1, p \rrbracket$, let $s_j$ be the endomorphism of $E_0$ induced by $R_j$. Show that $(s_1, \ldots, s_p)$ is an H-system of $E_0$. c) Deduce that $p(2m) \leqslant p(m) + 2$.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Show that if $n = 2^d m$ with $m$ odd, then $p(n) \leqslant 2d + 1$.

We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. Let $N = p(n)$ and $(a_1, \ldots, a_N)$ be an H-system of matrices of size $n$, that is, such that $$\forall i, a_i^2 = I_n \quad \text{and} \quad \forall i \neq j, a_i a_j + a_j a_i = 0$$ By considering the following matrices of $\mathcal{M}_{2n}(\mathbb{C})$ written in block form $$A_j = \left( \begin{array}{cc} a_j & 0 \\ 0 & -a_j \end{array} \right) (j \in \llbracket 1, N \rrbracket), \quad A_{N+1} = \left( \begin{array}{cc} 0 & I_n \\ I_n & 0 \end{array} \right), \quad A_{N+2} = \left( \begin{array}{cc} 0 & \mathrm{i} I_n \\ -\mathrm{i} I_n & 0 \end{array} \right)$$ show that $p(2n) \geqslant N + 2$.

Show that the matrices that are elements of $O ^ { + } ( 1,1 )$ are diagonalizable and find a matrix $P \in O ( 2 )$ such that, for every matrix $L \in O ^ { + } ( 1,1 )$, the matrix ${ } ^ { t } P L P$ is diagonal.

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 part $E$ is a real vector space of dimension $n$ equipped with a basis $\mathcal{B} = (\varepsilon_i)_{1 \leqslant i \leqslant n}$. We consider an endomorphism $f$ of $E$ and we denote by $A$ its matrix in the basis $\mathcal{B}$.
Determine thus the stable plane(s) of $f$ when $n = 3$ and $A$ is the matrix $A = \begin{pmatrix} 1 & -4 & 0 \\ 1 & -2 & -1 \\ 1 & 1 & 0 \end{pmatrix}$ considered in IV.F.

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum. Show that there exists an orthonormal basis $\left(v_{1}, \ldots, v_{n}\right)$ of $\mathbb{R}^{n}$ such that $$M = \sum_{i=1}^{n} m_{i} v_{i}\, {}^{t}v_{i}$$ Such a decomposition of $M$ will be called in the sequel a spectral resolution of $M$.

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Calculate $$\sup_{\|x\|=1} \langle x, Mx \rangle$$ as a function of the coordinates of $m$. Is this supremum attained? (One may decompose $x$ and $Mx$ on the orthonormal basis $\left(v_{1}, \ldots, v_{n}\right)$ of question 2a).

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$ its ordered spectrum, and $\left(v_{1}, \ldots, v_{n}\right)$ an orthonormal basis from the spectral resolution of $M$. Let $j$ be an integer, $1 \leqslant j \leqslant n$. We denote by $\mathcal{V}_{j}$ the vector subspace of $\mathbb{R}^{n}$ spanned by $\left(v_{1}, \ldots, v_{j}\right)$, and by $\mathcal{W}_{j}$ the one spanned by $\left(v_{j}, v_{j+1}, \ldots, v_{n}\right)$. Show the equalities $$\inf_{x \in \mathcal{V}_{j},\, \|x\|=1} \langle x, Mx \rangle = \sup_{x \in \mathcal{W}_{j},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$

Let $M \in \mathcal{S}_{n}(\mathbb{R})$, we denote by $m = s^{\downarrow}(M)$. Let $j$ be an integer, $1 \leqslant j \leqslant n$, and $\mathcal{V}$ be a vector subspace of $\mathbb{R}^{n}$ of dimension $j$. Show that $$\inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle \leqslant m_{j}.$$ (One may use questions $\mathbf{2c}$ and $\mathbf{3a}$, by choosing $\mathcal{U} = \mathcal{W}_{j}$.)

By using the notations of question 3b, deduce that $$\sup_{\mathcal{V} \subset \mathbb{R}^{n},\, \operatorname{dim} \mathcal{V} = j} \inf_{x \in \mathcal{V},\, \|x\|=1} \langle x, Mx \rangle = m_{j}.$$ Is this supremum attained?