TITLE: Why do siamese magic squares have real eigenvalues, symmetric around zero?
QUESTION [20 upvotes]: There is a standard method to construct magic squares of odd size, known as the 
 Siamese construction. I'll write $S_m$ for the $m \times m$ Siamese square. For example, here is $S_5$.
17    24     1     8    15
23     5     7    14    16
 4     6    13    20    22
10    12    19    21     3
11    18    25     2     9

As you can see, this matrix certainly isn't symmetric! It has one obvious eigenvector, the all ones vector, whose corresponding eigenvalue is $\frac{m^3+m}{2}$, and there is no obvious reason its other eigenvalues should have nice structure. Nonetheless, experimentation suggests:

(1) All eigenvalues of $S_m$ are real.
(2) If $\lambda$ is an eigenvalue of $S_m$ other than $\frac{m^3+m}{2}$, then $-\lambda$ is as well.

Can anyone explain why?
Remark In case anyone is curious how I found this, I am teaching myself MATLAB and came up with the task of listing characteristic polynomials of magic squares as a more or less random task which would involve learning some basic control structures and some mathematical tools. This piece of MATLAB documentation describes some alternate ways of thinking about the Siamese construction, which may be relevant.

REPLY [4 votes]: Here are some partial results.
This paper states that every regular magic square $\mathbf A\in\mathbb N^{n\times n}$ (that is, its entries satisfy $a_{i,j}+a_{n-i+1,n-j+1}=n^2+1$) is representable as a rank-$1$ correction of a skew-centrosymmetric matrix $\mathbf Z$ (that is, $\mathbf Z=-\mathbf J\mathbf Z\mathbf J$, where $\mathbf J$ is the exchange matrix, or the identity matrix with its columns reversed). In other words, the matrix you get when you subtract $\frac{n^2+1}{2}$ from all of the entries of $\mathbf A$ is skew-centrosymmetric. This is important since the eigenvalues of $\mathbf A$, apart from the magic eigenvalue $\frac{n^3+n}{2}$, are the same as the eigenvalues of $\mathbf Z$, which now has $0$ as an eigenvalue.
We can now use the results of this paper, which says that if $\lambda$ is an eigenvalue of $\mathbf Z$, then $-\bar{\lambda}$ must also be an eigenvalue, with the same algebraic multiplicity.
Thus, the remaining task is to prove that all the eigenvalues of a Siamese magic square are necessarily real.

A tantalizing result: Pressman, in this paper, shows the stronger result that if $\lambda$ is an eigenvalue of $\mathbf Z$, then $-\lambda$ is also an eigenvalue. There seems to be a criterion in the paper for saying when a skew-centrosymmetric matrix has all its eigenvalues real, but I have not yet figured how to apply it to this case.