Intuitive explanation of $S(p,w)\cdot p=0$ Can anyone provide an intuitive explanation of why the Slutsky matrix right multiplied by the price vector yields a zero matrix? 

I know this is true but I do not really understand why it is true. Can anyone help here?
This is a general mathematical property of the second derivative/Hessian matrix of multivariate functions that are homogeneous of degree one.  

The Expenditure function $E$ is homogeneous of degree one in prices. Why?
If all prices change in the same proportion (which is how we check for the mathematical property of homogeneity), relative prices do not change. If relative prices do not change, the quantitative composition of the minimum-cost compensated consumption bundle in order to achieve a given utility does not change at all. Then, since all prices have increased by the same proportion, budget shares remain the same, and the Expenditure needed to achieve the same utility, rises in that same proportion: homogeneity of degree one. 

By duality, the Hicksian demand vector is the gradient of the Expenditure function, $H = \nabla_p E$.

The Hicksian demand vector, gives us minimum-cost quantities demanded. Due to the homogeneity of degree one of the Expenditure function, the inner product of the Hicksian demand vector times the price vector equals the Expenditure function. This also should be intuitive: we just multiply each quantity demanded by the unit price that must be paid for it, and by summing these products we obtain the total Expenditure we must incur in order to acquire the minimum-cost bundle for given utility.

So we have  (simplifying differentiation notation) $ E = H\cdot p$ while also $\frac {\partial}{\partial p}E = H$. Therefore also 

$$\frac {\partial}{\partial p}(H\cdot p) = H \implies H + \frac {\partial H}{\partial p}\cdot p = H$$

and it must be the case that 

$$\frac {\partial H}{\partial p}\cdot p = 0$$

So the Hicksian demand vector is homogeneous of degree zero in prices (mathematically, this is a consequence of Euler's theorem for homogeneous functions, i.e. that if a function is homogeneous with degree of homogeneity $k$, its gradient has degree of homogeneity $k-1$).

But the 1st derivative (Jacobian) of Hicksian demand (which is the Hessian-matrix of second derivatives of the Expenditure function) is the Slutsky matrix, $\frac {\partial^2 E}{\partial p^2}=\frac {\partial H}{\partial p} = S(p,w)$. So $S(w,p) \cdot p = 0$.

So the result stems from the homogeneity of degree one of the Expenditure function. Is there an intuitive explanation, analogously with the intuition behind the homogeneity of degree one of the Expenditure function? Well, the former comes directly from the latter, so it is difficult to come up with a "separate" intuitive argument. One could informally say that compensated quantities demanded are "independent" of (not affected by) price variation when relative prices remain the same. Then in geometric terms, this means that the vectors of rates of change of compensated quantities demanded (which is what each row of the Slutsky matrix contains), are orthogonal to the price vector.