On the minimum number of Toeplitz factors of a matrix

TL;DR

This paper disproves a conjecture by Ye and Lim by showing certain 3x3 matrices cannot be factored into two Toeplitz matrices and improves bounds on the minimum number of Toeplitz factors needed for general matrices.

Contribution

It provides a counterexample to a previous conjecture and refines the bounds on Toeplitz factorization for small matrices.

Findings

Certain 3x3 matrices cannot be expressed as the product of two Toeplitz matrices.

Improved estimates for the minimum number of Toeplitz factors for small n.

Disproved a longstanding conjecture in matrix factorization theory.

Abstract

We disprove a conjecture by Ye and Lim, by showing that there are $3 \times 3$ complex matrices which can't be expressed as the product of two Toeplitz matrices of the same size. We also improve previous estimates by Ye and Lim on the minimum number of Toeplitz matrices needed to factor any $n \times n$ matrix, for low values of $n$.