Definition 1

A hermitian form $\Phi$ on $E$ is said to be positive (resp. negative) if $\Phi(x, x) \geqslant 0$ (resp. $\Phi(x, x) \leqslant 0$) for every $x \in E$.

When $A = K$, one also says that the quadratic form

associated with $\Phi$ is positive (resp. negative).

Suppose $E$ is finite-dimensional over $A$, and let $(e_i)$ ($i = 1, \ldots, n$) be an orthogonal basis of $E$ ($§ 6$, no 1, th. 1). For a hermitian form $\Phi$ on $E$ to be positive, it is necessary and sufficient that $\Phi(e_i, e_i) \geqslant 0$ for $i = 1, \ldots, n$. Let $\Phi$ be a positive non-degenerate hermitian form on $E$; since every positive element of $K$ is a square, hence of the form $\rho \overline{\rho}$ ($\rho \in A$), there exist orthonormal bases for $\Phi$ in $E$ ($§ 6$, no 1, cor. 1 of th. 1).