Exercise 7, § 3

Let $q(\xi_1, \ldots, \xi_n) = \sum_{i,j} a_{ij} \xi_i \xi_j$ ($a_{ij} = a_{ji}$) be a degenerate positive quadratic form on $\mathbf{R}^n$, such that $a_{ij} \leqslant 0$ for $i \neq j$. Assume that there is no partition $I \cup J$ of $\{1, 2, \ldots, n\}$ such that $a_{ij} = 0$ for $i \in I, j \in J$.

a) Show that, if we put $\xi_i = 0$, we obtain a non-degenerate positive quadratic form with respect to $\xi_1, \ldots, \xi_{i-1}, \xi_{i+1}, \ldots, \xi_n$.

b) Show that $a_{ii} > 0$ for all $i$. (Let $(\zeta_1, \ldots, \zeta_n)$ be an element of the kernel of $q$, with $\zeta_1 > 0, \ldots, \zeta_n > 0$. Use the equality $a_{1i} \zeta_1 + \cdots + a_{ni} \zeta_n = 0$.)

c) Show that if we replace one of the $a_{ij}$ by some $a'_{ij} < a_{ij}$, the new form is non-positive. (Use the equality $\sum_{i,j} a_{ij} \zeta_i \zeta_j = 0$.)