Gaussian quadrature formulae for matrix weights Original Research Article

We study Gaussian quadrature formulae for a matrix weight. We firstly show how to generate Gaussian quadrature formulae such that the sum of the rank of the quadrature matrix coefficients is a given nonnegative integer l: the nodes are the eigenvalues of certain perturbation of the truncated N-Jacobi matrix of size l or, equivalently, the zeros of certain matrix combination of two consecutive orthonormal polynomials, and the quadrature coefficients are the coefficients in the partial fraction decomposition of the ratio between the inverse of this orthonormal matrix polynomial and the associated polynomial of the second kind. We secondly prove that any Gaussian quadrature formula for a matrix weight must be of this form.