Every matrix is a product of Toeplitz matrices

TL;DR

This paper proves that any n-by-n matrix can be expressed as a product of a small number of Toeplitz or Hankel matrices, with specific bounds and computational methods, revealing unusual decomposition properties.

Contribution

It establishes the minimal number of Toeplitz or Hankel matrices needed to factorize any matrix and discusses the computational approaches for such decompositions.

Findings

Any matrix is generically a product of approximately n/2 Toeplitz matrices.

Decompositions into Toeplitz or Hankel matrices are generally not replaceable by arbitrary subspaces.

Such decompositions do not exist for symmetric Toeplitz, persymmetric Hankel, or circulant factors.

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.