Question about the relationship between Weak Axiom and Slutsky Matrix
It is almost true. 

There are examples of demand that have a negative definite Slutsky matrix but fails the Weak Axiom.

However, if we ask that
$$v \cdot S(p,w) v &lt;0 $$
whenever $v \not = \alpha p$ for any scalar $\alpha$ (i.e. $S$ is negative definite for all vectors except those proportional to price), then the Weak Axiom holds.