Cauchy matrix

Remarks

Example: with $x = (1, 2)$ and $y = (0, -1)$ the Cauchy matrix is $\begin{pmatrix} 1 & \tfrac12 \\ \tfrac12 & \tfrac13 \end{pmatrix}$; here $x_1 - y_2 = 2$ and $x_2 - y_2 = 3$. Non-example: $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ is not a Cauchy matrix, because no entry $1/(x_i - y_j)$ can be $0$. For $m = n$, the Cauchy determinant is $$\det C = \frac{\prod_{1 \le i < j \le n} (x_j - x_i)(y_i - y_j)}{\prod_{i=1}^n \prod_{j=1}^n (x_i - y_j)}.$$ So a square Cauchy matrix is invertible exactly when the $x_i$ are distinct and the $y_j$ are distinct. Every square submatrix of a Cauchy matrix is again a Cauchy matrix.