Matrix polynomials: Factorization via bisolvents

We reconsider the classification of all the factorizations of a matrix polynomial P as P = QR with Q a matrix polynomial and R(lambda) = lambda T - S a regular matrix pencil. It is shown that the entire classification problem can be reduced to the simpler classification of factors R with commuting coefficients S,T. It is then shown that, for these commuting factors, S and T must satisfy a certain algebraic equation which we call the bisolvent equation. This extends the generalized Bezout theorem which associates monic factors lambda I - S with solutions S of a solvent equation. In case P is regular, the classification of commuting pairs (S, T) of this type (up to left equivalence) is described in terms of enlarged standard pairs, following a well known approach. Under a non-derogatory generic condition on the roots of P, the number of such pairs associated with degree-minimal factorizations is finite, and admits explicit description in terms of Jordan chains. (C) 2017 Elsevier Inc. All rights reserved.