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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

xi = yi = i − ½.
Last updated: 05-27-2005 15:59:37
Science Fair Project Encyclopedia Contents Page
10-26-2009 08:16:03
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