A canonical Geronimus transformation for matrix orthogonal polynomials

We consider matrix polynomials orthogonal with respect to a sesquilinear form (Formula presented.) such that (Formula presented.) where µ 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 1. We obtain a connection formula between the sequences of matrix polynomials orthogonal with respect to [·, ·]W and µ, as well as a relation between the corresponding block Jacobi matrices. A non-symmetric sesquilinear form is also considered.