TITLE: Orthogonal Cauchy-like matrix
QUESTION [5 upvotes]: Given a $n \times n$ real Cauchy like matrix $C$, i.e., for real vectors $r$, $s$, $x$, $y$ 
$$
C_{ij} = \frac{r_i s_j}{ x_i - y_j}
$$

Can a Cauchy-like $C$ be orthogonal, i.e., $C C^T = I$ for $n > 2$?

There exists such an orthogonal $C$ for $n = 2$ , $x = [1,0.4]$, $y = [6.25,0.625]$, $r = [-1.8114, 1.4811]$, and $s = [2.3367, -0.1225]$ with 
$$
C = \begin{bmatrix}
0.8062   & 0.5916 \\
-0.5916  &  0.8062
\end{bmatrix}
$$

REPLY [4 votes]: Please give a look at this short note.