# Cauchy matrix

Let $x_{1}$, $x_{2},\ldots,x_{m}$, and $y_{1}$, $y_{2}\ldots,y_{n}$ be elements in a field $F$, satisfying the that

1. 1.

$x_{1},\ldots,x_{m}$ are distinct,

2. 2.

$y_{1},\ldots,y_{n}$ are distinct, and

3. 3.

$x_{i}+y_{j}\neq 0$ for $1\leq i\leq m$, $1\leq j\leq n$.

The matrix

 $\begin{bmatrix}\frac{1}{x_{1}+y_{1}}&\frac{1}{x_{1}+y_{2}}&\cdots&\frac{1}{x_{% 1}+y_{n}}\\ \frac{1}{x_{2}+y_{1}}&\frac{1}{x_{2}+y_{2}}&\cdots&\frac{1}{x_{2}+y_{n}}\\ \vdots&\vdots&\ddots&\vdots\\ \frac{1}{x_{m}+y_{1}}&\frac{1}{x_{m}+y_{2}}&\cdots&\frac{1}{x_{m}+y_{n}}\end{bmatrix}$

is called a Cauchy matrix over $F$.

The determinant of a square Cauchy matrix is

 $\frac{\prod_{i

Since $x_{i}$’s are distinct and $y_{j}$’s are distinct by definition, a square Cauchy matrix is non-singular. Any submatrix of a rectangular Cauchy matrix has full rank.

Title Cauchy matrix CauchyMatrix 2013-03-22 14:30:43 2013-03-22 14:30:43 kshum (5987) kshum (5987) 9 kshum (5987) Definition msc 15A57 Cauchy matrices