Cauchy Matrix - Cauchy Determinants

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).

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

    (Schechter 1959, Theorem 1)

where A_i(x) and B_i(x) are the Lagrange polynomials for and, respectively. That is,

with