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
\begin{equation}
S(p,w)=D_px(p,w)+D_wx(p,w)x(p,w)^T
\end{equation}
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?
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.