Restricted inversion of split-Bezoutians

Abstract.

The main aim of the present paper is to compute inverses of split-Bezoutians considered as linear operators restricted to subspaces of symmetric or skewsymmetric vectors. Such results are important, e.g., for the inversion of nonsingular, centrosymmetric or centroskewsymmetric Toeplitz-plus-Hankel Bezoutians B of order n. To realize this inversion we present algorithms with O(n²) computational complexity, which involves an explicit representation of B^–1 as a sum of a Toeplitz and a Hankel matrix. Based on different ideas such inversion formulas have already been proved in previous papers by the authors. Here we focus on the occurring splitting parts since they are of interest also in a more general context. The main key is the solution of the converse problem: the inversion of Toeplitz-plus-Hankel matrices. An advantage of this approach is that all appearing special cases can be dealt with in the same, relatively straightforward way without any additional assumptions.

In memory of Georg Heinig