General Christoffel Perturbations for Mixed Multiple Orthogonal Polynomials

Performing both right and left multiplication operations using general regular matrix polynomials, which need not be monic and may possess leading coefficients of arbitrary rank, on a rectangular matrix of measures associated with mixed multiple orthogonal polynomials, reveals corresponding Christoffel formulas. These formulas express the perturbed mixed multiple orthogonal polynomials in relation to the original ones. Utilizing the divisibility theorem for matrix polynomials, we establish a criterion for the existence of perturbed orthogonality, expressed through the non-cancellation of certain $\tau$ determinants.