# Question about the relationship between Weak Axiom and Slutsky Matrix

We know that if a differentiable Walrasian demand function $$x(p,w)$$ satisfies Walras' law ($$p^Tx=w$$), homogeneity of degree zero ($$x(\alpha p,\alpha w)=x(p,w)$$), and the weak axiom of revealed preference, then at any $$(p,w)$$, the Slutsky matrix $$$$S(p,w)=D_px(p,w)+D_wx(p,w)x(p,w)^T$$$$ is negative semidefinite.

My question is: if $$S(p,w)$$ is negative semidefinite, then what can we say about the demand function $$x(p,w)$$? Can we conclude that $$x(p,w)$$ satisfies weak axiom?

However, if we ask that $$v \cdot S(p,w) v <0$$ 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.