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A note on generalized companion pencils in the monomial basis

Abstract

In this paper, we introduce a new notion of generalized companion pencils for scalar polynomials over an arbitrary field expressed in the monomial basis. Our definition is quite general and extends the notions of companion pencil in De Terán et al. (Linear Algebra Appl 459:264–333, 2014), generalized companion matrix in Garnett et al. (Linear Algebra Appl 498:360–365, 2016), and Ma–Zhan companion matrices in Ma and Zhan (Linear Algebra Appl 438: 621–625, 2013), as well as the class of quasi-sparse companion pencils introduced in De Terán and Hernando (INdAM Series, Springer, Berlin, pp 157–179, 2019). We analyze some algebraic properties of generalized companion pencils. We determine their Smith canonical form and we prove that they are all nonderogatory. In the last part of the work we will pay attention to the sparsity of these constructions. In particular, by imposing some natural conditions on its entries, we determine the smallest number of nonzero entries of a generalized companion pencil.

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Acknowledgements

We are very much indebted to an anonymous referee for a very careful reading of the manuscript and for many helpful suggestions that allowed us to improve significantly the original version. We also thank a second referee for suggesting the inclusion of several references on companion matrices for polynomials expressed in other bases than the monomial basis, like [5, 8, 33, 36, 37].

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Correspondence to Fernando De Terán.

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This work has been partially supported by the Ministerio de Economía y Competitividad of Spain through Grants MTM2017-90682-REDT and MTM2015-65798-P.

An earlier version of this paper was presented at the Conference “Linear Algebra, Matrix Analysis and Applications. ALAMA2018”, held in Sant Joan d’Alacant on May/June 2018.

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De Terán, F., Hernando, C. A note on generalized companion pencils in the monomial basis. RACSAM 114, 8 (2020). https://doi.org/10.1007/s13398-019-00760-y

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Keywords

  • Companion matrix
  • Companion pencil
  • Linearization
  • Sparsity
  • Scalar polynomial
  • Matrix polynomial
  • Arbitrary field
  • Digraph
  • Composite cycle
  • Extension field
  • Ring of polynomials
  • Field of fractions

Mathematics Subject Classification

  • 15A22
  • 65F15
  • 05C20
  • 05C50
  • 15B99