Schur complements of Bezoutians and the inversion of block Hankel and block Toeplitz matrices

Abstract

We derive a closed inversion formula for an np × np square block Hankel matrix H n − 1 = ( W i − j ) with entries W i from the ring of the p × p matrices over a field. The representation of H −1 n − 1 relies upon a strong structure-preserving property of the Schur complements of the nonsingular leading principal submatrices of a certain generalized Bezoutian of matrix polynomials. These polynomials may be determined by computing iteratively the Schur complements of the nonsingular leading principal submatrices of H n − 1 in an augmented block Hankel matrix H ̃ which is a Bezoutian matrix up to multiplication by proper reverse matrices. Since the special structure of the Bezoutian matrix is preserved at any step of the computation, such method can be efficiently implemented at an overall computational cost of O( p 3 n 2) operations in the average case and O( p 3 n 3) operations in the worst case. In the normal case, a suitable modification which uses both divide and conquer strategies and fast polynomial arithmetic reaches the bound of O( p 3 n log 2 n) arithmetical operations.