Cauchy matrix

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In mathematics, the Cauchy matrix is an m×n matrix A, whose elements are given by

$a_{ij}={\frac{1}{x_i+y_j}};\quad x_i+y_j\neq 0,\quad 1 \le i \le m,\quad 1 \le j \le n.\,$

where x_i and y_j are elements of a field $\mathcal{F}$ , and where (x_i) and (y_j) are injective sequences (they do not contain repeated elements; elements are distinct).

[edit] Properties

When m=n and the matrix is square, the determinant, known as a Cauchy determinant, is given explicitly by

$\det A={{\prod_{i<j} (x_i-x_j)\prod_{i<j} (y_i-y_j)}\over {\prod_{i,j} (x_i +y_j)}}.\,$

As a consequence of the injectivity of $(x i)$ and $(y j)$ , all square Cauchy matrices are invertible.
Every submatrix of a Cauchy matrix is itself a Cauchy matrix.