TITLE: Showing that a certain matrix is not positive definite
QUESTION [18 upvotes]: 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?

REPLY [17 votes]: 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 $(x-1)^6 (x-\frac{10}{3})$.