Generalized Bezoutian and the inversion problem for block matrices, I. General scheme

Abstract

The unified approach to the matrix inversion problem initiated in this work is based on the concept of the generalized Bezoutian for several matrix polynomials introduced earlier by the authors. The inverse X-1 of a given block matrix X is shown to generate a set of matrix polynomials satisfying certain conditions and such that X-1 coincides with the Bezoutian associated with that set. Thus the inversion of X is reduced to determining the underlying set of polynomials. This approach provides a fruitful tool for obtaining new results as well as an adequate interpretation of the known ones. © 1986 Birkhäuser Verlag.