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.



What is known about the spectrum of a Cauchy matrix?
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.](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.81.3678&rep=rep1&type=pdf)
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](http://books.google.com/books?id=H8fwqfqACdEC&lpg=PA92&ots=dNykJhjse4&dq=totally%20positive%20cauchy%20matrix&pg=PA92#v=onepage&q=totally%20positive%20cauchy%20matrix&f=false).


