Let $h \in \mathscr{L}(E)$ be an endomorphism of $E$. Show that the following properties are equivalent: i) $h$ is an element of $\operatorname{Sim}(E)$; ii) $h^{*}h$ is collinear to $\operatorname{Id}_{E}$; iii) the matrix of $h$ in an orthonormal basis of $E$ is collinear to an orthogonal matrix.
Let $f$ be an antisymmetric endomorphism of $E$. Show that, if $S$ is a vector subspace of $E$ stable under $f$, then $S^{\perp}$ is stable under $f$. Show that the endomorphisms induced by $f$ on $S$ and on $S^{\perp}$ are antisymmetric.
Let $f$ be an antisymmetric endomorphism of $E$. Let $g$ be an antisymmetric endomorphism of $E$, such that $fg = -gf$. Show that: $\forall x \in E, \langle f(x), g(x) \rangle = 0$.
Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$. We fix $x \in E \backslash \{0\}$. By considering $\Phi : f \mapsto f(x)$, linear application from $V$ to $E$, show that $\operatorname{dim}(V) \leqslant n$. Thus $1 \leqslant d_{n} \leqslant n$.
In this question only, we assume $n = 2$. Explicitly give a vector space of dimension 2, formed of similarity matrices. Deduce from this, carefully, that $d_{2} = 2$.
In this question only, we assume $n$ is odd. If $f, g$ belong to $GL(E)$, show that there exists $\lambda \in \mathbb{R}$ such that $f + \lambda g$ is non-invertible. One may reason by considering the characteristic polynomial of $fg^{-1}$. Deduce that $d_{n} = 1$.
Let $V$ be a vector subspace of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of dimension $d \geqslant 1$. Show that there exists a vector subspace $W$ of $\mathscr{L}(E)$ included in $\operatorname{Sim}(E)$, of the same dimension $d$, and containing $\operatorname{Id}_{E}$.
Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that for all $i \in \{1, 2, \ldots, d-1\}$, $f_{i}^{*} + f_{i}$ is collinear to $\operatorname{Id}_{E}$.
Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. Let $\left(\operatorname{Id}_{E}, f_{1}, \ldots, f_{d-1}\right)$ be a basis of $V$. Show that there exists a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ such that for all $i \in \{1, 2, \ldots, d-1\}$, $g_{i}$ is antisymmetric (one will seek $g_{i}$ as a combination of $f_{i}$ and $\operatorname{Id}_{E}$).
Let $V$ be a vector subspace of $\mathscr{L}(E)$ containing $\operatorname{Id}_{E}$, included in $\operatorname{Sim}(E)$ and of dimension $d \geqslant 2$. We fix a basis $\left(\operatorname{Id}_{E}, g_{1}, \ldots, g_{d-1}\right)$ of $V$ with for all $i$, $g_{i}$ antisymmetric. a) Show that for all $i \neq j$, $g_{i}g_{j} + g_{j}g_{i}$ is collinear to $\operatorname{Id}_{E}$. b) Show that we define a scalar product on $\mathscr{L}(E)$ by setting, for all $f, g$ of $\mathscr{L}(E)$: $(f \mid g) = \operatorname{tr}(f^{*}g)$. We consider, in the rest of this question, a basis $(h_{1}, \ldots, h_{d-1})$ of $\operatorname{Vect}(g_{1}, \ldots, g_{d-1})$ orthogonal for this scalar product. c) Show that the $h_{i}$ are antisymmetric and satisfy: $\forall i \neq j, h_{i}h_{j} + h_{j}h_{i} = 0$. What should be done so that the $h_{i}$ are also orthogonal automorphisms?
Conversely, let $(h_{1}, \ldots, h_{d-1})$ be a family of $\mathscr{L}(E)$ such that the $h_{i}$ are antisymmetric orthogonal automorphisms satisfying for all $i \neq j$, $h_{i}h_{j} + h_{j}h_{i} = 0$. Show that $\operatorname{Vect}\left(\operatorname{Id}_{E}, h_{1}, \ldots, h_{d-1}\right)$ is a vector subspace of $\mathscr{L}(E)$, of dimension $d$, included in $\operatorname{Sim}(E)$.
Let $p$ be an odd integer such that $\operatorname{dim}(E) = n = 2p$. We assume that there exist $d \geqslant 3$ and a family $(f_{1}, f_{2}, \ldots, f_{d-1})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let $x \in E$ of norm 1. a) Show that $(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is an orthonormal family, and that $S = \operatorname{Vect}(x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is stable under $f_{1}$ and $f_{2}$. b) Deduce that $d_{n-4} \geqslant 3$.
In this section, the dimension of $E$ is 4. We assume that there exists a vector subspace of $\mathscr{L}(E)$ of dimension 4 included in $\operatorname{Sim}(E)$. We then consider, in accordance with I.D.4, a family $(f_{1}, f_{2}, f_{3})$ of elements of $\mathscr{L}(E)$ such that the $f_{i}$ are orthogonal automorphisms, antisymmetric satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Let a fixed vector $x \in E$ of norm 1. a) Justify that the family $B = (x, f_{1}(x), f_{2}(x), f_{1}f_{2}(x))$ is a basis of $E$ then show that there exist real numbers $\alpha, \beta, \gamma, \delta$ such that: $$f_{3}(x) = \alpha x + \beta f_{1}(x) + \gamma f_{2}(x) + \delta f_{1}f_{2}(x)$$ Show that $\alpha = \beta = \gamma = 0$ and that $\delta \in \{-1, 1\}$. b) Show that $f_{3} = \delta f_{1}f_{2}$. If necessary, by replacing $f_{3}$ with its opposite, we assume in what follows that $f_{3} = f_{1}f_{2}$. c) If $x_{0}, x_{1}, x_{2}, x_{3}$ are real numbers, give the matrix $M(x_{0}, x_{1}, x_{2}, x_{3})$ in $B$ of the endomorphism $x_{0}\operatorname{Id}_{E} + x_{1}f_{1} + x_{2}f_{2} + x_{3}f_{3}$.
In this section, the dimension of $E$ is 4. Verify that for all $(x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4}$, $M(x_{0}, x_{1}, x_{2}, x_{3})$ is a similarity matrix. What can we conclude?
In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. By using $f_{4}$, show that $f_{3}$ cannot be equal to $\pm f_{1}f_{2}$.
In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. Show that $f_{1}f_{2}f_{3}$ is an orthogonal automorphism, symmetric and not collinear to $\operatorname{Id}_{E}$.
In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. What is the spectrum of $f_{1}f_{2}f_{3}$? Show that there exists $x \in E$ of norm 1 such that $\langle f_{1}f_{2}f_{3}(x), x \rangle = 0$. We fix such an $x$ for the rest.
In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. We fix $x \in E$ of norm 1 such that $\langle f_{1}f_{2}f_{3}(x), x \rangle = 0$. Show that $F = (x, f_{1}(x), f_{2}(x), f_{3}(x), f_{1}f_{2}(x), f_{1}f_{3}(x), f_{2}f_{3}(x), f_{1}f_{2}f_{3}(x))$ is an orthonormal family.
In this section, the dimension of $E$ is 12. We assume that there exists in $\mathscr{L}(E)$, a family $(f_{1}, f_{2}, f_{3}, f_{4})$ of antisymmetric orthogonal automorphisms satisfying: $\forall i \neq j, f_{i}f_{j} + f_{j}f_{i} = 0$. We set $V = \operatorname{Vect}(F)$ where $F = (x, f_{1}(x), f_{2}(x), f_{3}(x), f_{1}f_{2}(x), f_{1}f_{3}(x), f_{2}f_{3}(x), f_{1}f_{2}f_{3}(x))$. It is therefore a vector subspace of $E$ of dimension 8. a) Show that $V^{\perp}$ is stable under $f_{1}, f_{2}, f_{3}$. b) We denote by $f_{i}^{\prime}$ the endomorphism induced by $f_{i}$ on $V^{\perp}$, $i = 1, 2, 3$. Justify that there exists $\delta^{\prime} \in \{-1, 1\}$ such that $f_{3}^{\prime} = \delta^{\prime} f_{1}^{\prime} f_{2}^{\prime}$. If necessary, by replacing $f_{3}$ by $-f_{3}$, we consider for the rest that $f_{3}^{\prime} = f_{1}^{\prime} f_{2}^{\prime}$. c) Let $e$ be fixed in $V^{\perp}$, of norm 1. By proceeding as in II.B.1.a) (but this is not to be redone), one can show that $(e, f_{1}(e), f_{2}(e), f_{1}f_{2}(e))$ is an orthonormal basis of $V^{\perp}$. By noting that $f_{3}(e) = f_{1}f_{2}(e)$, use this basis to show that: $\forall y \in V^{\perp}, f_{4}(y) \in V$. Thus $W = f_{4}(V^{\perp})$ is a vector subspace of $V$ of dimension 4. d) Show that the sum of $W$ and $V^{\perp}$ is direct and that $W \oplus V^{\perp}$ is stable under $f_{1}, f_{2}, f_{3}, f_{4}$. Then reach a contradiction.