Remark 7

We suppose that $A$ satisfies the conditions at the beginning of § 7 and that $E$ is of finite dimension $n$. Let $\Phi$ be a non-degenerate positive hermitian form on $E$. a) Now prove that every similitude for $\Phi$ can be written in one and only one way as the product of a homothety of ratio $\geqslant 0$ in $K$, and of a unitary transformation. b) For every basis $(a_i)_{1 \leq i \leq n}$ of $E$, show that there exists one and only one orthonormal basis $(e_i)_{1 \leq i \leq n}$ of $E$ satisfying the following conditions: 1° for every $m$ such that $1 \leq m \leq n$, the subspace generated by $a_1, \ldots, a_m$ is identical with the subspace generated by $e_1, \ldots, e_m$; 2° one has $\Phi(a_i, e_i) \geq 0$ in $K$, for every index $i$ (cf. § 6, no 1, prop. 1). c) Deduce from b) that for every invertible square matrix $M$ of order $n$ over $A$, there exists one and only one pair of matrices $(L, U)$ of order $n$, such that $U$ is a unitary matrix, that $L = (\lambda_{ij})$ has only zeros below its diagonal and diagonal terms $\lambda_{ii}$ belonging to $K$ and $> 0$, and finally that $M = LU$.