Cauchy matrix

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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[edit]

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).

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,

with

Generalization[edit]

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

  • approximate Cauchy matrix-vector multiplication with ops (e.g. the fast multipole method),
  • (pivoted) LU factorization with ops (GKO algorithm), and thus linear system solving,
  • approximated or unstable algorithms for linear system solving in .

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[edit]

References[edit]