linear algebra - Are hollow positive symmetric matrices negative definite on subspace $sum_i x_i=0$?

I am interested in the properties of symmetric hollow matrices (diagonal full of zeros) for which all off-diagonal terms are strictly positive. An example for $n=3$ would be

$$\left[\begin{array}{ccc} 0 & A_{21} & A_{31}\\ A_{21} & 0 & A_{32}\\ A_{31} & A_{32} & 0 \end{array}\right]$$
where $A_{21}$, $A_{31}$ and $A_{32}$ are all strictly positive. Let's call the set of these matrices $B$.

Now consider the subspace $S:\sum_i x_i=0$. I want to show that matrices in $B$ are negative definite on $S$. I.e. for all $x\in S$ and for a matrix $A\in B$ we have $x'Ax\leq -d \left\Vert x\right\Vert$ for some $d>0$. This seems to be the case for $n=2$ and $n=3$ but I would like a general proof. If the result does not hold, could we show that matrices in $B$ are negative semi definite on $S$ instead?

Answer

Not true. For example, with $n=3$ take $$x = \pmatrix{1\cr 1\cr-2\cr}$$ with $A_{21}$ very large. $A_{21} > 2 A_{31} + 2 A_{32}$ will do.