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Given a matrix polynomial $W(x)$, matrix bi-orthogonal polynomials withrespect to the sesquilinear form $\langle P(x),Q(x)\rangle_W=\int P(x)W(x)\operatorname{d}\mu(x)(Q(x))^{\top}$, $P(x),Q(x)\in\mathbb R^{p\timesp}[x]$, where $\mu(x)$ is a matrix of Borel measures supported in some infinitesubset of the real line, are considered. Connection formulas between thesequences of matrix bi-orthogonal polynomials with respect to $\langle\cdot,\cdot\rangle_W$ and matrix polynomials orthogonal with respect to$\mu(x)$ are presented. In particular, for the case of nonsingular leadingcoefficients of the perturbation matrix polynomial $W(x)$ we present ageneralization of the Christoffel formula constructed in terms of the Jordanchains of $W(x)$. For perturbations with a singular leading coefficient severalexamples by Dur\'an et al are revisited. Finally, we extend these results tothe non-Abelian 2D Toda lattice hierarchy.

Christoffel transformations for matrix orthogonal polynomials in the real line and the non-Abelian 2D Toda lattice hierarchy