An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line

We consider matrix polynomials orthogonal with respect to a sesquilinear form <., .>(W), such that < P(t)W(t), Q(t)W(t)>(W) = integral(J) P(t)d(mu)Q(t)(T), P, Q is an element of P-pxp [t], where mu is a symmetric, positive definite matrix of measures supported in some infinite subset J of the real line, and W(t) is a matrix polynomial of degree N. We deduce the integral representation of such sesquilinear forms in such a way that a Sobolev-type inner product appears. We obtain a connection formula between the sequences of matrix polynomials orthogonal with respect to mu and <., .>(W), as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.

matrix orthogonal polynomials; block jacobi matrices; matrix geronimus transformation; block cholesky decomposition; block lu decomposition; quasi-determinants; differential-equations

An Extension of the Geronimus Transformation for Orthogonal Matrix Polynomials on the Real Line Articles