Foundations of Computational Mathematics

, Volume 16, Issue 3, pp 577–598 | Cite as

Every Matrix is a Product of Toeplitz Matrices

Article
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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.

Keywords

Toeplitz decomposition Hankel decomposition Linear algebraic geometry 

Mathematics Subject Classification

14A10 15A23 15B05 20G20 65F30 
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Notes

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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Copyright information

© SFoCM 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA
  2. 2.Computational and Applied Mathematics Initiative, Department of StatisticsUniversity of ChicagoChicagoUSA

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