properties of $\begin{bmatrix}-A& -B^T\\ -B &0\end{bmatrix}$

Question

Consider the following matrix $$C=\begin{bmatrix}-A& -B^T\\ -B &0\end{bmatrix}$$ where $A>0$ and B is a matrix such that the diagonal entries of B are all zero and the rest of the entries are either zero or 1, and $(I-B) \mathbf{1}=0$ and $\mathbf{1}^T(I-B)=0$, where $\mathbf{1}$ is the vector of all ones). Do the eigenvalues of C have special properties?

Answer 1

Here are some obvious properties (let $A$ be $n\times n$):

1. As $C$ is real symmetric, all its eigenvalues are real.
2. The characteristic polynomial of $C$ is $\det(\lambda^2I + \lambda A - B^TB).$
3. As $A$ is positive definite and the Schur complement of $A$ is $BA^{-1}B^T$, the nullitiy of $C$ is equal to the nullity of $B$. The null space of $C$ consists of vectors of the form $(0,v^T)^T$, where $B^Tv=0$.
4. $C$ is indefinite. For any $v\in\mathbb{R}^n$, $(\mathbf{1}^T,v^T)C(\mathbf{1}^T,v^T)^T =-\mathbf{1}^TA\mathbf{1}-2(\mathbf{1}^Tv)$. Therefore, one can choose $v$ to make the quantity positive or negative at will.