Cauchy determinant

In mathematics, the Cauchy determinant in linear algebra, named after Augustin Cauchy, is the determinant of the complex n×n matrix CM with entries

$a_{ij}={1\over {(x_i+y_j)}}$ for $1 \le i,j \le n.\,$

Here it is assumed that

$x_i+y_j \ne 0\;\; \forall\; i,j.\,$

The explicit formula for the determinant is

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

Example

The determinant of the Hilbert matrix is the case

x_i = y_i = i − ½.

Last updated: 05-27-2005 15:59:37

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