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.
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
- 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).