Properties of matrix for which $x^TCx=0\forall x\geq0$

Question

Consider two matrices $A$ and $B$ and a vector $x$ with $x\geq0$, i.e. the vector $x$ lies in the nonnegative orthant.

What properties does the resulting matrix $C=A^TB$ have if

$$x^TCx=0\forall x\geq0$$

I vaguely remember that this type of quadratic form had a certain name, though I may be wrong. Please let me know if you know.

Answer 1

Let $S$ be the symmetric part of $C$. Then $x^TSx=\frac12\left[x^TCx+(x^TCx)^T\right]=0$ for all $x\ge0$. It follows that $$ x^TSy=\frac12\left[(x+y)^TS(x+y)+(x-y)^TS(x-y)-x^TSx-y^TSy\right]=0 $$ for every $x,y\ge0$. In particular, if we take $x=e_i$ and $y=e_j$, we have $s_{ij}=0$ for every $(i,j)$. Hence $S=0$ and $C$ must be skew-symmetric.