Every matrix is a product of Toeplitz matrices

Abstract

We show that every n-by-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-by-n matrices. Furthermore such decompositions do not exist if we require the factors to be symmetric Toeplitz, persymmetric Hankel, or circulant matrices, even if we allow an infinite number of factors. Lastly, we discuss how the Toeplitz and Hankel decompositions of a generic matrix may be computed by either (i) solving a system of linear and quadratic equations if the number of factors is required to be [n/2] + 1, or (ii) Gaussian elimination in O(n^3) time if the number of factors is allowed to be 2n.