Orthogonal Matrix Polynomials: Zeros and Blumenthal's Theorem

Abstract

In this paper, we establish a quadrature formula and some basic properties of the zeros of a sequence (Pn)nof orthogonal matrix polynomials on the real line with respect to a positive definite matrix of measures. Using these results, we show how to get an orthogonalizing matrix of measures for a sequence (Pn)nsatisfying a matrix three-term recurrence relation. We prove Blumenthal's theorem for orthogonal matrix polynomials describing the support of the orthogonalizing matrix of measures in case the matrix recurrence coefficients associated with these matrix polynomials tend to matrix limits having the same entries on every diagonal.