We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that $$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$ Let $(e_1, \ldots, e_n)$ be the canonical basis of $\mathbb{R}^n$ and, for $i \in \llbracket 1, n \rrbracket$, let $u_i$ be the endomorphism of $\mathbb{R}^n$ defined by $\forall X \in \mathbb{R}^n, u_i(X) = B(X, e_i)$. The matrix of $u_i$ in the canonical basis of $\mathbb{R}^n$ will be denoted $A_i$. We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$. For $j = 1, \ldots, n-1$ we denote by $S_j$ the complex matrix $S_j = \mathrm{i} {}^t A_n A_j$. a) Prove that $(S_1, \ldots, S_{n-1})$ is an H-system. b) Deduce that we have the inequality $p(n) \geqslant n - 1$ where $p(n)$ is defined in section I.C.
We assume that $n \geqslant 3$ and that there exists a bilinear map $B$ such that
$$\forall X, Y \in \mathbb{R}^n, \quad \|X\| \times \|Y\| = \|B(X, Y)\|$$
Let $(e_1, \ldots, e_n)$ be the canonical basis of $\mathbb{R}^n$ and, for $i \in \llbracket 1, n \rrbracket$, let $u_i$ be the endomorphism of $\mathbb{R}^n$ defined by $\forall X \in \mathbb{R}^n, u_i(X) = B(X, e_i)$. The matrix of $u_i$ in the canonical basis of $\mathbb{R}^n$ will be denoted $A_i$. We denote by $p(n)$ the largest integer $p \geqslant 1$ such that $E$ admits an H-system of cardinality $p$.\\
For $j = 1, \ldots, n-1$ we denote by $S_j$ the complex matrix $S_j = \mathrm{i} {}^t A_n A_j$.\\
a) Prove that $(S_1, \ldots, S_{n-1})$ is an H-system.\\
b) Deduce that we have the inequality $p(n) \geqslant n - 1$ where $p(n)$ is defined in section I.C.