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

Given a matrix polynomial $W(x)$, matrix bi-orthogonal polynomials with respect to the sesquilinear form $\langle P(x),Q(x)\rangle_W=\int P(x) W(x)\operatorname{d}μ(x)(Q(x))^{\top}$, $P(x),Q(x)\in\mathbb R^{p\times p}[x]$, where $μ(x)$ 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 $\langle \cdot,\cdot\rangle_W$ and matrix polynomials orthogonal with respect to $μ(x)$ 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 et al are revisited. Finally, we extend these results to the non-Abelian 2D Toda lattice hierarchy.

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