Remark 4

One assumes that $A = K$ or $A = K(i)$. Let $R$ be a hermitian matrix of order $n$ over $A$; one says that $R$ has signature $(s, t)$ if the hermitian form on $A^n$ having $R$ as matrix with respect to the canonical basis, has signature $(s, t)$. Let $\Delta_k$ denote the principal minor of $R$ obtained by deleting in $R$ the rows and columns of index $> k$; one assumes that $\Delta_{s+t} \neq 0$ and that, for no index $k < s + t$, $\Delta_k$ and $\Delta_{k+1}$ are simultaneously zero (cf. § 6, exerc. 1 b)). Show that if $\Delta_k = 0$ for some $k < s + t$, $\Delta_{k-1}$ and $\Delta_{k+1}$ have opposite signs (method of exerc. 1 a) of § 6) and that the number $s - t$ is equal to

(use exerc. 2 of § 6).