Cauchy matrix

Summary

A Cauchy matrix has entries $1/(x_i - y_j)$ for two lists of numbers $x_1, \dots, x_m$ and $y_1, \dots, y_n$ with $x_i \ne y_j$. Its determinant has an explicit product formula, the Cauchy determinant. As a result, a square Cauchy matrix is invertible when the $x_i$ are distinct and the $y_j$ are distinct. The Hilbert matrix is the best known Cauchy matrix.