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 (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 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

    (Schechter 1959, eqn 4; Cauchy 1841, p. 154, eqn. 10).

It is always nonzero, and thus all square Cauchy matrices are invertible. The inverse A−1 = B = [bij] is given by

    (Schechter 1959, Theorem 1)

where Ai(x) and Bi(x) are the Lagrange polynomials for and , respectively. That is,



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 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 denotes the size of the matrix (one usually deals with square matrices, though all algorithms can be easily generalized to rectangular matrices).

See also