Corollary 2

— Let $(a_{ij})_{1 \leq i,j \leq n}$ be a hermitian matrix, with complex elements, and with determinant D. Suppose that the inequality

holds for all complex numbers $z_1, ..., z_n$. Then

Suppose D is non-null; the equality $D = a_{11} ... a_{nn}$ holds if and only if $a_{ij} = 0$ for all $i \neq j$.

Let $\Phi$ be the hermitian form on the vector space $\mathbf{C}^n$ given by

for $\mathbf{z} = (z_1, ..., z_n)$ and $\mathbf{z}' = (z'_1, ..., z'_n)$ in $\mathbf{C}^n$. By hypothesis, $\Phi$ is positive.

First assume that $\Phi$ is separating, that is, that D is non-null. If $(e_1, ..., e_n)$ is the canonical basis of $\mathbf{C}^n$, we have $\Phi(e_i, e_j) = a_{ij}$, and cor. 2 follows immediately from cor. 1, a) by putting $x_i = e_i$.

Since $a_{ii} = \Phi(e_i, e_i) \geq 0$, we also have inequality (40) if $D = 0$.