Eigevalues of block matrix of order $n$ What are the eigenvalues of the following block matrix?
$$\begin{bmatrix} J_2-I_2 & J_2 & J_2-I_2 &  J_2-I_2 &\cdots & J_2\\ J_2 & J_2-I_2 & J_2 & J_2-I_2 &\cdots &J_2-I_2 \\ J_2-I_2 & J_2 & J_2-I_2 &J_2 & \cdots & J_2-I_2 \\ J_2-I_2 & J_2-I_2 & J_2 & J_2-I_2 & \cdots & J_2-I_2 \\ \vdots & \vdots & \vdots & \vdots &  \ddots &  \vdots  \\J_2 & J_2-I_2 & J_2-I_2 & J_2-I_2 & \cdots & J_2-I_2 \end{bmatrix}_{n}$$ 
Here, $J_2$ is a matrix of order 2 whose all values are $1$s and $I_2$ is an identity matrix of order 2.
 A: Note: In the current version of the question, we have $p = 2$.
It looks like your matrix can be written in the form 
$A = J_n \otimes (J_p - I) + B\otimes I$
where $\otimes$ denotes the Kroncker product and $B$ is the $n \times n$ matrix
$$
B = \pmatrix{0&1&0 && \cdots & 1\\
1&0&1&0&\cdots&0\\
0&1&0&\ddots&\cdots &\vdots\\
\vdots&&\ddots&\ddots&&0\\
0&\cdots &&&0&1\\
1&0&\cdots&0&1&0}
$$
Note, however, that the two matrices being added commute.  It suffices, then, to find the eigenvalues and eigenvectors of $J_n, (J_p - I), B,$ and $I$ separately, and then use the properties of the Kronecker product and commuting matrices.

In fact, following the hint in the comment below: let $F_k$ denote the $k \times k$ DFT matrix.  We find that
$$
(F_n \otimes F_p)^*[J_n \otimes (J_p - I) + B\otimes I](F_n \otimes F_p) = \\
(F_n^*J_n F_n) \otimes ([F_p^*J_p F_p] - I) +
(F_n^*B F_n) \otimes I
$$
where all matrix products in parentheses produce diagonal matrices.  Thus, we have diagonalized $A$, and all that remains is to find the diagonal entries in the above computation.
It is also notable that the eigenvalues of the $n \times n$ and $p \times p$ matrices are simply the eigenvalues of circulant matrices, as described here.
For example, the eigenvalues of $B$ are $2\cos(2 \pi j/n)$ for $j = 1,\dots,n$ (and $F_n^*BF_n$ has those diagonal values in that order).
