On the minimum number of Toeplitz factors of a matrix

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 $n$.