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.

$x_{1},\ldots,x_{m}$ are distinct,
2.

$y_{1},\ldots,y_{n}$ are distinct, and
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<j}(x_{i}-x_{j})(y_{i}-y_{j})}{\prod_{ij}(x_{i}+y_{j})}

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
Canonical name	CauchyMatrix
Date of creation	2013-03-22 14:30:43
Last modified on	2013-03-22 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