Eigenvalues of centrosymmetric matrix I made some number of calculations with doubly centrosymmetric matrices $ 4 \times 4$ with positive integer entries (by doubly centrosymmetric I mean matrix which stays the same after rotation its entries about central point by $90^\circ$ - example below)  and I've received interesting results for its eigenvalues: 
matrix has always (if inverse exists) two real positive eigenvalues and two pure imaginary. 
Additionally these pure imaginary are always integer whereas real ones usually are not integer.  


*

*How these facts can be explained ?


Example:
$\begin{bmatrix}
23 & 15 & 17 & 23 \\
17  & 53 & 53 & 15 \\
15  & 53 & 53 & 17  \\
23 & 17 & 15 & 23  \end{bmatrix}$
Eigenvalues:
$(119.863,  0.000i),
( 32.137,  0.000i),
(  0.000,  2.000i),
(  0.000, -2.000i)$
Especially, it is interesting why pure imaginary are integer in contrast to real ones?
 A: Nice observation!  We can confirm that your observation holds (except that one of the real roots may be negative) for all $4\times 4$ matrices of the form you specified by analyzing the characteristic polynomial of the matrix.
Let's denote the elements of the matrix satisfying your form as
$A=\begin{bmatrix}
a & b & c & a \\
c  & d & d & b \\
b  & d & d & c  \\
a & c & b & a  \end{bmatrix}.$
The matrix has only four degrees of freedom, so it's no surprise that the characteristic polynomial $p(\lambda)=\det(A-\lambda I)$ can be written down fairly compactly.  After computing, simplifying, and factoring the characteristic polynomial, we get
$$p(\lambda)=\{\lambda^2-2(a+d)\lambda+4ad-(b+c)^2\}\times\{\lambda + (b-c)^2\}.$$
For $a,b,c,d\in\mathbb{Z^+},$ this gives pure imaginary solutions $\pm|(b-c)|i$ and real solutions $(a+d)\pm\sqrt{(a-d)^2+(b+c)^2}.$
As a check, we can look at your example with $a=23,b=15,c=17,$ and $d=53.$ Plugging in these values into the roots we found, we see that your example matrix has eigenvalues $\pm 2i,$ $76+2\sqrt{481},$ and $76-2\sqrt{481}.$
