Every Matrix is a Product of Toeplitz Matrices

TLDR: In this article, it was shown that toeplitz and Hankel matrices do not have a subspace of size at most 2n+5, and that such subspaces do not exist even if the factors are symmetric Toplitz or persymmetric Hankel.

Abstract: We show that every $$n\,\times \,n$$n×n matrix is generically a product of $$\lfloor n/2 \rfloor + 1$$?n/2?+1 Toeplitz matrices and always a product of at most $$2n+5$$2n+5 Toeplitz matrices. The same result holds true if the word `Toeplitz' is replaced by `Hankel,' and the generic bound $$\lfloor n/2 \rfloor + 1$$?n/2?+1 is sharp. We will see that these decompositions into Toeplitz or Hankel factors are unusual: We may not, in general, replace the subspace of Toeplitz or Hankel matrices by an arbitrary $$(2n-1)$$(2n-1)-dimensional subspace of $${n\,\times \,n}$$n×n matrices. Furthermore, such decompositions do not exist if we require the factors to be symmetric Toeplitz or persymmetric Hankel, even if we allow an infinite number of factors.