A Cauchy matrix 
is a matrix whose entries have the form
 |
where 
for all indices in the matrix. An equivalent convention
uses 
by changing the signs of the second parameter set (Higham 1995). For an 
Cauchy matrix, its determinant
is
 |
This is the Cauchy determinant formula (Krattenthaler 1999, p. 6). For a square matrix, distinct 
and distinct 
, together with the nonzero-denominator condition, give a
nonsingular matrix (Pan 2001).
The Hilbert matrix is the example 
, 
. Cauchy matrices are an important family of structured
matrices.
