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  "body": "\n## Definition\n\nLet $F$ be a [[field|04C5ZBG1WT]], and let $x_1, \\dots, x_m$ and $y_1, \\dots, y_n$ be elements of $F$ with $x_i \\ne y_j$ for all $i, j$.\nThe *Cauchy matrix* of these elements is the $m \\times n$ [[matrix|0VD9ZSWWJG]] $C$ with entries\n$$C_{ij} = \\frac{1}{x_i - y_j} \\qquad (1 \\le i \\le m,\\ 1 \\le j \\le n).$$\n\n## Remarks\n\nExample: with $x = (1, 2)$ and $y = (0, -1)$ the Cauchy matrix is $\\begin{pmatrix} 1 \u0026 \\tfrac12 \\\\ \\tfrac12 \u0026 \\tfrac13 \\end{pmatrix}$; here $x_1 - y_2 = 2$ and $x_2 - y_2 = 3$.\nNon-example: $\\begin{pmatrix} 1 \u0026 0 \\\\ 0 \u0026 1 \\end{pmatrix}$ is not a Cauchy matrix, because no entry $1/(x_i - y_j)$ can be $0$.\nFor $m = n$, the Cauchy determinant is\n$$\\det C = \\frac{\\prod_{1 \\le i \u003c j \\le n} (x_j - x_i)(y_i - y_j)}{\\prod_{i=1}^n \\prod_{j=1}^n (x_i - y_j)}.$$\nSo a square Cauchy matrix is invertible exactly when the $x_i$ are distinct and the $y_j$ are distinct.\nEvery square submatrix of a Cauchy matrix is again a Cauchy matrix.\n",
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