Proposition 3

*Suppose E finite-dimensional and A commutative ($\mathbf{A} = \mathbf{K}$ or $\mathbf{A} = \mathbf{K}(i)$). Let $\Phi$ be a hermitian form on E, X its matrix with respect to a basis $(x_j)$ ($j = 1, \ldots, n$) of E; for every subset H of $\{1, n\}$, denote by $X_{H, H}$ the minor of X obtained by deleting the rows and columns with indices $j \notin H$ (Chap. III, § 6, No. 3).

a) *If $\Phi$ is positive non-degenerate, then $X_{H, H} > 0$ for every subset H of $\{1, n\}$.

b) *Conversely, if, on putting $H_j = \{1, j\}$, we have $X_{H_j, H_j} > 0$ for $j = 1, \ldots, n$, then $\Phi$ is positive non-degenerate.

Suppose first that $\Phi$ is positive non-degenerate. The elements $(x_j)$ ($j \in H$) form a basis of a subspace F of E, and the minor $X_{H, H}$ is the discriminant of the restriction $\Phi_F$ of $\Phi$ to F with respect to this basis; now, since $\Phi_F(x, x) > 0$ for every $x \neq 0$ in F, $\Phi_F$ is positive non-degenerate (Corollary to Proposition 2); hence $X_{H, H} > 0$ (Proposition 1). To prove b), let us note that, with the notation of Proposition 1, § 6, No. 1, the minor $X_{H_j, H_j}$ is equal to $D_{j+1, j+1}$; there therefore exists (Proposition 1, § 6, No. 1) an orthogonal basis $(e_j)$ ($j = 1, \ldots, n$) of E such that $\Phi(e_j, e_j) > 0$ for $j = 1, \ldots, n$; consequently $\Phi$ is positive non-degenerate.