Explicit formulas for the inverses of Toeplitz matrices, with applications

TL;DR

This paper presents explicit formulas for inverting truncated block Toeplitz matrices related to multivariate stationary processes, enabling efficient computation and analysis of such systems.

Contribution

It introduces novel explicit formulas for the inverses of block Toeplitz matrices using Fourier coefficients, applicable to multivariate processes and ARMA models.

Findings

Provides a strong convergence result for block Toeplitz systems.

Derives closed-form inverses for multivariate ARMA process matrices.

Enables linear-time algorithms for solving block Toeplitz systems.

Abstract

We derive novel explicit formulas for the inverses of truncated block Toeplitz matrices that correspond to a multivariate minimal stationary process. The main ingredients of the formulas are the Fourier coefficients of the phase function attached to the spectral density of the process. The derivation of the formulas is based on a recently developed finite prediction theory applied to the dual process of the stationary process. We illustrate the usefulness of the formulas by two applications. The first one is a strong convergence result for solutions of general block Toeplitz systems for a multivariate short-memory process. The second application is closed-form formulas for the inverses of truncated block Toeplitz matrices corresponding to a multivariate ARMA process. The significance of the latter is that they provide us with a linear-time algorithm to compute the solutions of corresponding block Toeplitz systems.