Exercise 8, § 3

Let $(a_{ij})$ be a real symmetric matrix with $n$ rows and $n$ columns.

a) Put $s_k = \sum_{i=1}^n a_{ik}$. Then, for all $\xi_1, \ldots, \xi_n \in \mathbf{R}$,

b) Let $\zeta_1, \ldots, \zeta_n \in \mathbf{R}^*$. Put $\sum_i \zeta_i a_{ik} = t_k$. Then

(In a), replace $\xi_i$ by $\frac{\xi_i}{\zeta_i}$ and $a_{ik}$ by $\zeta_i \zeta_k a_{ik}$.)

c) If there exist numbers $\zeta_1, \ldots, \zeta_n > 0$ such that $\sum_i \zeta_i a_{ik} = 0$ ($k = 1, 2, \ldots, n$), and if $a_{ij} \leqslant 0$ for $i \neq j$, then the quadratic form $\sum_{i,k} a_{ik} \xi_i \xi_k$ is degenerate and positive. (Use b).)

d) Let $\sum_{i,j} q_{ij} \xi_i \xi_j$ be a quadratic form on $\mathbf{R}^n$ such that $q_{ij} \leqslant 0$ for $i \neq j$. Assume that there is no partition $I \cup J$ of $\{1, 2, \ldots, n\}$ such that $q_{ij} = 0$ for $i \in I, j \in J$. Then the form is non-degenerate and positive if and only if there exist $\zeta_1 > 0, \ldots, \zeta_n > 0$ such that $\sum_i \zeta_i q_{ik} = 0$ ($k = 1, \ldots, n$).