A Cauchy matrix is one with entries $a_{ij} = \frac{1}{x_i \pm y_j}$.
A Cauchy matrix, is an $m\times n$ matrix with elements $a_{ij}$ in the form
$$a_{ij}=\frac{1}{x_i-y_j};\quad x_i-y_j\neq 0,\quad 1\leq i\leq m,\quad 1\leq j\leq n$$
where $x_i$ and $y_j$ are elements of a field $\mathbb{F}$, and $(x_i)$ and $(y_j)$ are injective sequences.
The Hilbert matrix is a special case of the Cauchy matrix, where
$x_{i}-y_{j}=i+j-1.$
Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
