Cauchy matrix
Let ${x}_{1}$, ${x}_{2},\mathrm{\dots},{x}_{m}$, and ${y}_{1}$, ${y}_{2}\mathrm{\dots},{y}_{n}$ be elements in a field $F$, satisfying the that

1.
${x}_{1},\mathrm{\dots},{x}_{m}$ are distinct,

2.
${y}_{1},\mathrm{\dots},{y}_{n}$ are distinct, and

3.
${x}_{i}+{y}_{j}\ne 0$ for $1\le i\le m$, $1\le j\le n$.
The matrix
$$\left[\begin{array}{cccc}\hfill \frac{1}{{x}_{1}+{y}_{1}}\hfill & \hfill \frac{1}{{x}_{1}+{y}_{2}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \frac{1}{{x}_{1}+{y}_{n}}\hfill \\ \hfill \frac{1}{{x}_{2}+{y}_{1}}\hfill & \hfill \frac{1}{{x}_{2}+{y}_{2}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \frac{1}{{x}_{2}+{y}_{n}}\hfill \\ \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\vdots}\hfill & \hfill \mathrm{\ddots}\hfill & \hfill \mathrm{\vdots}\hfill \\ \hfill \frac{1}{{x}_{m}+{y}_{1}}\hfill & \hfill \frac{1}{{x}_{m}+{y}_{2}}\hfill & \hfill \mathrm{\cdots}\hfill & \hfill \frac{1}{{x}_{m}+{y}_{n}}\hfill \end{array}\right]$$ 
is called a Cauchy matrix over $F$.
The determinant^{} of a square Cauchy matrix is
$$ 
Since ${x}_{i}$’s are distinct and ${y}_{j}$’s are distinct by definition, a square Cauchy matrix is nonsingular. Any submatrix^{} of a rectangular Cauchy matrix has full rank.
Title  Cauchy matrix 

Canonical name  CauchyMatrix 
Date of creation  20130322 14:30:43 
Last modified on  20130322 14:30:43 
Owner  kshum (5987) 
Last modified by  kshum (5987) 
Numerical id  9 
Author  kshum (5987) 
Entry type  Definition 
Classification  msc 15A57 
Defines  Cauchy matrices 