What are the eigenvalues of a particular type of block partitioned matrix

Question

Let $C=\begin{bmatrix} A & \pm J \\ \pm J^T & B\end{bmatrix}$ be a square block partitioned matrix of order $m+n$ where $A$ and $B$ are square symmetric matrices of orders $m$ and $n$ respectively with entries from the set $\{-1,0,1\}$ such that the diagonal entries of $A$ and $B$ are always zero and $J$ is a matrix of all ones. Can anyone suggest method to find the characteristic polynomial and eigenvalues of $C$ in terms of the corresponding facts of $A$ and $B$ ( and if required that of $J$ also) ?

Answer 1

For the case where both plus signs occur in $C$:

Shur's formula when $\lambda$ is not an eigenvalue of $A$ gives $$ {\rm det\,}\begin{bmatrix} A-\lambda I & J \cr J & B-\lambda I \end{bmatrix} = {\rm det\,}(A-\lambda I){\rm det\,}(-J(A-\lambda I)^{-1}J+B-\lambda I). $$ For any matrix $M$ you have that $JMJ = kJ$ where $k$ is the sum of all elements of $M$. Therefore $$ {\rm det\,}C = {\rm det\,}(A-\lambda I){\rm det\,}(B-a(\lambda)J-\lambda I), $$ where $a(\lambda)$ is the sum of all elements of $(A-\lambda I)^{-1}$.