Cauchy matrix


In mathematics, a Cauchy matrix, named after Augustin Louis Cauchy, is an m×n matrix with elements aij in the form
where and are elements of a field, and and are injective sequences.
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 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 Ai and Bi 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
Here denotes the size of the matrix.