Every Matrix is a Product of Toeplitz Matrices

We show that every $$n,times ,n$$ matrix is generically a product of $$lfloor n/2 rfloor + 1$$ toeplitz matrices and always a product of at most $$2n+5$$ toeplitz matrices. the same result holds true if the word ‘toeplitz’ is replaced by ‘hankel,’ and the generic bound $$lfloor n/2 rfloor + 1$$ is sharp. we will see that these decompositions into toeplitz or hankel factors are unusual: we not, in general, replace the subspace of toeplitz or hankel matrices by an arbitrary $$(2n-1)$$ -dimensional subspace of $${n,times ,n}$$ 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.