Linear Algebra Trivia: Can anyone identify this class of matrix? Consider a matrix:
\begin{pmatrix}
 0 & -y & x \\
 x & 0 & -y \\
 -y & x & 0 \\
\end{pmatrix}
where $x,y$ are positive real numbers
I wish to identy the most "specific" class of matrices that the above matrix belongs to. I had tought that this matrix was skew symmetric at first, but clearly it is not because it is not symmetric UNLESS $x = y$, then it is real skew symmetric. 
But what about the general case? Is there a special name for this type of matrix? I am thinking something along the line of circular...circulant...Toeplitz...
 A: It's circulant, with constant diagonal from left-to-right.  
Which means you can then use convenient formulas (specific to circulant matrices) to compute its eigenvalues.
(It does not have constant diagonal from right-to-left, so I do not believe it is another form of a matrix other than circulant.)
A: What you have is  $xA - y A^2,$ where $A$ is
\begin{pmatrix}
 0 & 0 & 1 \\
 1 & 0 & 0 \\
 0 & 1 & 0 \\
\end{pmatrix}
The set of all $wI + x A + y A^2$ makes a commutative ring, actually an algebra; the point being it is closed under multiplication. The determinant is $w^3 + x^3 + y^3 - 3wxy.$ Forcing the variables to be integers, we find that the product of two numbers that can be expressed as $w^3 + x^3 + y^3 - 3wxy$ is a third number that can be represented as $w^3 + x^3 + y^3 - 3wxy.$ I have not checked carefully, this is sort of a degenerate form, I think the set of integrally represented numbers has only some $3$-adic restriction. Yes, evidently the only restriction is, with $n = w^3 + x^3 + y^3 - 3wxy,$ we have $n \neq \pm 3 \pmod 9.$
