Every Matrix is a Product of Toeplitz Matrices

We show that every n × n matrix is generically a product of n/2 + 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 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)-dimensional subspace of n × n matrices. Furthermore, suchdecompositions donot exist ifwe require the factors to be symmetric Toeplitz or persymmetric Hankel, even if we allow an infinite number of factors.