Exercise 6, § 3

a) Let K be a commutative field, V a vector space of finite dimension n over K, $\varphi$ a symmetric bilinear form on V, and N the kernel of $\varphi$. Assume that $\dim N = 1$. Show that the kernel of the extension of $\varphi$ to $\bigwedge^{n-1} V$ is of dimension $n-1$.

b) Assume in addition that $K = \mathbf{R}$ and that $\varphi$ is positive. Let $(e_1, \ldots, e_n)$ be a basis of V, and $a_{ij} = \varphi(e_i, e_j)$. Assume that $a_{ij} \leq 0$ for $i \neq j$. Assume that there is no partition $I \cup J$ of $\{1, 2, \ldots, n\}$ such that $a_{ij} = 0$ for $i \in I, j \in J$. Let $A_{ij}$ be the cofactor of $a_{ij}$ in the matrix $(a_{ij})$. Show that $A_{ij} > 0$ for all $i$ and $j$. (Let $\zeta_1 e_1 + \cdots + \zeta_n e_n$ be a vector generating N, all of whose coordinates are $> 0$. Show that every row and column of $(A_{ij})$ is proportional to $(\zeta_1, \ldots, \zeta_n)$.

Deduce that $A_{ij} = \mu \zeta_i \zeta_j$ for some constant $\mu$. By considering the $A_{ii}$, show that $\mu > 0$.

c) Show that $\zeta_1, \ldots, \zeta_n$ are proportional to $\sqrt{A_{11}}, \ldots, \sqrt{A_{nn}}$.