Block matrix of order $2n$ related How to find eigenvalues of following block matrix?
$$M=\begin{bmatrix}J-A & J-A \\J-A & J\end{bmatrix}$$
Where $A$ is any $n \times n$ matrix and $J$ is $n \times n$ matrix whose all entries are $1$
 A: Lemma: A,B,C,D$\in \mathbb{M}_{n\times n}$. If AC=CA, then $$\begin{vmatrix}A&B\\C&D\end{vmatrix}=\begin{vmatrix}AD-CB\end{vmatrix}$$
Thus, in this problem
$$
\begin{vmatrix}\lambda I_{2n}-M\end{vmatrix}
=\begin{vmatrix}\lambda I_n-J+A&A-J\\A-J&\lambda I_n-J\end{vmatrix}
=\begin{vmatrix}(\lambda I_n-J+A)(\lambda I_n-J)-(A-J)^2\end{vmatrix}
=\begin{vmatrix}\lambda^2 I_n-2\lambda J+\lambda A-A^2+JA\end{vmatrix}
$$
The solution of the equation
$$\begin{vmatrix}\lambda^2 I_n-2\lambda J+\lambda A-A^2+JA\end{vmatrix}=0$$
is the eigenvalue that we want.
A: The matrix $M$ can be expressed as 
$$M=\begin{bmatrix}J-A & J-A \\J-A & 0\end{bmatrix} + \begin{bmatrix}0 & 0 \\0 & J\end{bmatrix} = \begin{bmatrix}1 & 1 \\1 & 0\end{bmatrix}\otimes (J-A) + ff' \equiv K\otimes (J-A) + ff'$$
where $f = [0_{1,n}, j_{1,n}]^T$ and $j_{1,n}$ is the matrix of size $1\times n$ with all elements equal 1.
Further, let $U_K$, $T_K$ be the Schur factorization of the matrix $K$ and $U_A$, $T_A$ be the Schur factorization of the matrix $J-A$. Then
$$M= (U_KT_KU_K')\otimes (U_AT_AU_A') + ff' = (U_K\otimes U_A)(T_K\otimes T_A)(U_K\otimes U_A)'+ ff'$$
Thus, the Schur factorization of $K\otimes (J-A)$ can be obtained from the Schur factorization of $K$ and $J-A$. Finally, the Schur factorizationof $M$ is given by rank-1 update of the Schur factorization of $K\otimes (J-A)$.
If $A$ is symmetric, then finding eigenvalues of $M$, knowing the Schur factorization of $A$ can be done very efficiently. Observe, that the Schur factorization of $J-A$ is again rank-1 update of the Schur factorization $A$. Rank-1 update of the spectral decoposition of symmetric matrix is a standard tool and can be performed in $O(n^2)$ operations.
For nonsymmetric matrix $A$ rank-1 updates stil can be performed, but are much more complicated.
