In mathematics, a Cauchy matrix, named after Augustin-Louis Cauchy, is an m×n matrix with elements aij in the form where x_i and y_j are elements of a field \mathcal{F}, and (x_i) and (y_j) are injective sequences (they contain distinct elements).

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 (x_i) and (y_j). If the sequences were not injective, the determinant would vanish, and tends to infinity if some x_i tends to y_j. 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 = [bij] is given by where Ai(x) and Bi(x) are the Lagrange polynomials for (x_i) and (y_j), respectively. That is, with

Generalization

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