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.
$x_{i}+y_{j}\eq 0$ 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.