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 and are elements of a field, and and are injective sequences.

Properties

Every submatrix of a Cauchy matrix is itself a Cauchy matrix.
The Hilbert matrix is a special case of the Cauchy matrix, where

Cauchy determinants

The determinant of a Cauchy matrix is clearly a rational fraction in the parameters and. If the sequences were not injective, the determinant would vanish, and tends to infinity if some tends to. 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 = is given by
where A_i and B_i are the Lagrange polynomials for and, respectively. That is,
with

Generalization

A matrix C is called Cauchy-like if it is of the form
Defining X=diag, Y=diag, one sees that both Cauchy and Cauchy-like matrices satisfy the displacement equation
. 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

approximate Cauchy matrix-vector multiplication with ops,
LU factorization with ops, and thus linear system solving,
linear system solving in ops with the use of fast matrix multiplication algorithms, instead of ops without it, where is the displacement rank and.
approximated or unstable algorithms for linear system solving in.

Here denotes the size of the matrix.