Cauchy matrix

In mathematics, a Cauchy matrix, named after Augustin Louis Cauchy, is an "m"×"n" matrix with elements "a""ij" in the form
where formula_2 and formula_3 are elements of a field formula_4, and formula_5 and formula_6 are injective sequences (they contain "distinct" elements).
The Hilbert matrix is a special case of the Cauchy matrix, where
Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
Cauchy determinants.
The determinant of a Cauchy matrix is clearly a rational fraction in the parameters formula_5 and formula_6. If the sequences were not injective, the determinant would vanish, and tends to infinity if some formula_2 tends to formula_3. A subset of its zeros and poles are thus known. The fact is that there are no more zeros and poles:
The determinant of a square Cauchy matrix A is known as a Cauchy determinant and can be given explicitly as
It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A−1 = B = [bij] is given by 
where "A"i(x) and "B"i(x) are the Lagrange polynomials for formula_5 and formula_6, respectively. That is, 
with
Generalization.
A matrix C is called Cauchy-like if it is of the form
Defining X=diag(xi), Y=diag(yi), one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation
(with formula_20 for the Cauchy one). Hence Cauchy-like matrices have a common displacement structure, which can be exploited while working with the matrix. For example, there are known algorithms in literature for
Here formula_24 denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).