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
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 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).