On Orthogonal Matrix Polynomials

In this paper we deal with orthogonal matrix polynomials. First of all, we establish some basic notations and results we need later. A matrix polynomial P is a matrix whose entries are polynomials, or, equivalently, a combination P(t) = A 0 +A 1 t+ +A n t n , where A 0 ; ; A n are numerical matrices (the size of all the matrices which appear in this paper is N N). A positive deenite matrix of measures W is a matrix whose entries are Borel measures and such that for any Borel set A 2 R, the numerical matrix W(A) is positive semideenite. A sequence of matrix polynomials (P n) n , dgr(P n) = n, is orthonormal with respect to a positive deenite matrix of measures W if: Z P n (t)dW (t)P m (t) = n;m Id: Matrix orthonormality is equivalent to the following three-term matrix recurrence relation tP n (t) = A n+1 P n+1 (t) + B n P n (t) + A n P n?1 (t); with initial condition P ?1 = and P 0 a nonsingular numerical matrix, and where the B n 's are hermitian and the A n 's are nonsingular (see 1, 4, 8]). Using the polar decomposition of a matrix , the A n 's can be taken to be positive deenite, while the QR decomposition of Francis and Kublanovskaja allows us to take the A n 's to be lower triangular matrices. We can associate to the sequence of orthogonal matrix polynomials (P n) n the (2N + 1)-banded innnite Hermitian matrix which is called the N-Jacobi matrix As in the scalar case, the polynomials of the second kind are deened as Q n (z) = Z P n (z) ? P n (t) z ? t dW(t): The sequence (Q n) n satisses the same three-term recurrence relation as that satissed by (P n) n but with initial conditions Q 0 = , Q 1 a nonsingular numerical matrix. The paper is divided in two parts. In the rst one we show a close link between scalar polyno-mials satisfying higher order recurrence relations and orthogonal matrix polynomials. Using this relationship, we give a nontrivial interpretation of scalar orthogonality as matrix orthogonality, and, as a consequence, we will see that orthogonal matrix polynomials are going to be a useful tool to study some problems in the classical …