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⟩=∫_IP(t)dμQ(t)^T,P,Q∈P^p×p[t],where μ is a symmetric, positive definite matrix of measures supported in some infinite subset I 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 μ and ⟨ · , · ⟩ _W, as well as a relation between the corresponding block Jacobi and Hessenberg type matrices.