## Inversion of centrosymmetric Toeplitz-plus-Hankel Bezoutians

Torsten Ehrhardt and Karla Rost

### Abstract

In this paper we discuss how to compute the inverse of a nonsingular, centrosymmetric Toeplitz-plus-Hankel Bezoutian $B$ of order $n$ and how to find a representation of $B^{-1}$ as a sum of a Toeplitz and a Hankel matrix. Besides the known splitting property of $B$ as a sum of two split-Bezoutians, the connection of the latter to Hankel Bezoutians of about half size is used. The fast inversion of the Hankel Bezoutians together with an inversion formula, which was the subject of a previous paper, leads us to an inversion formula for $B^{-1}$ as a Toeplitz-plus-Hankel matrix. It also enables us to design an $O(n^2)$ inversion algorithm.

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### Key words

Bezoutian matrix, Toeplitz matrix, Hankel matrix, Toeplitz-plus-Hankel matrix, matrix inversion

### AMS subject classifications

15A09, 15B05, 65F05

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