Lemma: Let $A=(a_{ij})$ and $B=(b_{ij})$ be two positive semi-definite matrices, i.e. $$\sum_{i,j=1}^{n} a_{ij} x_{i}x_{j} \geq 0$$ for all $x=(x_1,...,x_n) \in R^n$, similarly for B. Then $$\sum_{i,j=1}^{n}a_{ij}b_{ij} \geq 0$$.
Based on above lemma I want to show that: if $A=(a_{ij})$ is a symmetric positive matrix ($A \geq 0$) and $B=(b_{ij})$ is non-positive symetric matrix ($B \leq 0$) then $$\sum_{i,j=1}^{n}a_{ij}b_{ij} \leq 0$$.
Thank you very much for help.