Orthogonal matrix polynomials: zeros and Blumenthal’ Theorem

Abstract. In this paper, we establish a quadrature formula and some basic properties of the zeros of a sequence (Pn)n of orthogonal matrix polynomials on the real line with respect to a positive de¯nite matrix of measures. Using these results, we show how to get an orthog-onalizing matrix of measures for a sequence (Pn)n satisfying 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 coe±cients associated with these matrix polynomials tend to matrix limits having the same entries on every diagonal. 1. INTRODUCTION. A close relationship between orthogonal matrix polynomials (or matrix polynomials satisfying a matrix three-term recurrence formula) and scalar polynomials satisfying a higher order recurrence formula has been established very recently (see [D1, D2 and DV]). Thus, in [DV], the following theorem has been prove