Let $J\_k$ be a $k \times k$ all ones matrix and $B$ **any** $k \times k$ binary matrix - that is $B$ only has entries from $\{0,1\}$.


I would like to show that the matrix $$X\_B = (J\_k -I) - B (J\_k - I)^{-1} B^T\,,$$ is not positive-definite. In other words, I'd like to show that 



> 
> At least one eigenvalue of $X\_B$ is non-positive. 
> 
> 
> 


I can show that for certain specific matrices $B$ but don't see how to prove the more general statement. Does anybody know why this property seem to hold in general?



Showing that a certain matrix is not positive definite
Counterexample: let $k=7$, and let $B$ be the circulant matrix with $B\_{ij}=1$ **iff** $i-j \in \{1,2,4\} \bmod 7$. Then $X\_B$ is $I + \frac12 J$, with characteristic polynomial $(x-1)^6 (x-\frac92)$. Or use $B+I$ instead to get $I + \frac13 J$, with characteristic polynomial $(x-1)^6 (x-\frac{10}{3})$.


