Math people:
A Cauchy matrix is an $m$-by-$n$ matrix $A$ whose elements have the form
$a_{i,j} = \frac{1}{x_i-y_j}$, with $x_i \neq y_j$ for all $(i, j)$, and the $x_i$'s and $y_i$'s belong to a field (http://en.wikipedia.org/wiki/Cauchy_matrix). Also it seems to be part of the definition that the $x_i$'s and $y_j$'s are all distinct (does anyone know why?). I am only interested in the case where the field is the real numbers, and all the $x_i$'s and $y_j$'s are positive integers. My question is, what is known about the eigenvalues of a square, real Cauchy matrix? There is a formula for its determinant, which gives you their product, and the trace of the matrix, which is their sum, is easy to find. I have Googled this extensively and found almost nothing.
I originally posted this on Math Stack Exchange but I got no answers so I removed the question and I am posting it here.
Best Answer
Suppose $x_i > 0$ and $y_j -x_j$, then $c_{ij} = 1/(x_i+x_j)$. These matrices are infinitely divisible, i.e., $[c_{ij}^r]$ is also positive definite for all $r > 0$.
Spectral properties of Cauchy-like matrices and kernels are studied here. For additional information and an easier read, on the general case ($1/(x_i+y_j)$), you might enjoy looking at the recent book by Pinkus.