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Given a  matrix polynomial   $W(x)$  , matrix bi-orthogonal polynomials with respect to the sesquilinear form    $$ \begin{align*} {\left\langle{P(x),Q(x)}\right\rangle}_{W}=\int P(x)W(x)\text{d}\mu(x) (Q(x))^{\top}, \quad P,Q\in{\mathbb{R}}^{p\times p}[x], \end{align*} $$where   $\mu(x)$   is a matrix of Borel measures supported in some infinite subset of the real line, are considered. Connection formulas between the sequences of matrix bi-orthogonal polynomials with respect to   ${\left\langle{\cdot, \cdot }\right\rangle}_W$   and matrix polynomials orthogonal with respect to   $\mu(x)$   are presented. In particular, for the case of nonsingular leading coefficients of the perturbation matrix polynomial   $W(x)$   we present a generalization of the  Christoffel formula constructed in terms of the Jordan chains of   $W(x)$  . For perturbations with a singular leading coefficient, several examples by Durán and coworkers are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy.

Christoffel Transformations for Matrix Orthogonal Polynomials in the Real Line and the non-Abelian 2D Toda Lattice Hierarchy