A fast elementary algorithm for computing the determinant of toeplitz matrices

TL;DR

This paper presents a new elementary proof for a fast algorithm to compute the determinant of Toeplitz matrices, leveraging the eigenvalues of associated companion matrices, with explicit formulas for small bandwidths.

Contribution

It introduces a simplified proof of a known fast determinant computation method for Toeplitz matrices and provides explicit formulas for small bandwidth cases.

Findings

The determinant can be expressed via the eigenvalues of a related companion matrix.

The paper offers symbolic formulas for small bandwidth Toeplitz matrices.

Provides examples illustrating the elementary proof and formulas.

Abstract

In recent years, a number of fast algorithms for computing the determinant of a Toeplitz matrix were developed. The fastest algorithm we know so far is of order $k^2\log{n}+k^3$, where $n$ is the number of rows of the Toeplitz matrix and $k$ is the bandwidth size. This is possible because such a determinant can be expressed as the determinant of certain parts of $n$-th power of a related $k \times k$ companion matrix. In this paper, we give a new elementary proof of this fact, and provide various examples. We give symbolic formulas for the determinants of Toeplitz matrices in terms of the eigenvalues of the corresponding companion matrices when $k$ is small.