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Every Matrix is a Product of Toeplitz Matrices

Abstract

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 may 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.

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Notes

  1. We restrict our attention to decompositions that exist for arbitrary matrices over both \(\mathbb {R}\) and \(\mathbb {C}\). Of the six decompositions described in [47], we discounted the Cholesky (only for positive definite matrices), Schur (only over \(\mathbb {C}\)), and spectral decompositions (only for normal matrices).

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Acknowledgments

We thank Professor T. Y. Lam for inspiring this work. This article is dedicated to his 70th birthday. We would also like to thank the anonymous referees for their invaluable comments, particularly for the argument after Corollary 1 that substantially simplifies our deduction of Hankel decomposition from Toeplitz decomposition. LHL’s work is partially supported by AFOSR FA9550-13-1-0133, NSF DMS 1209136, and NSF DMS 1057064. KY’s work is partially supported by NSF DMS 1057064 and NSF CCF 1017760.

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Correspondence to Lek-Heng Lim.

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Communicated by Nicholas Higham.

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Ye, K., Lim, LH. Every Matrix is a Product of Toeplitz Matrices. Found Comput Math 16, 577–598 (2016). https://doi.org/10.1007/s10208-015-9254-z

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Keywords

  • Toeplitz decomposition
  • Hankel decomposition
  • Linear algebraic geometry

Mathematics Subject Classification

  • 14A10
  • 15A23
  • 15B05
  • 20G20
  • 65F30