Skip to main content

A note on generalized companion pencils in the monomial basis


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.

This is a preview of subscription content, access via your institution.


  1. 1.

    Antoniou, E.N., Vologiannidis, S.: A new family of companion forms for polynomial matrices. Electron. J. Linear Algebra 11, 78–87 (2004)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Aurentz, J., Mach, T., Vandebril, R., Watkins, D.S.: Fast and backward stable computation of roots of polynomials. SIAM J. Matrix Anal. Appl. 36, 942–973 (2015)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Aurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.S.: Fast and backward stable computation of roots of polynomials, part II: backward error analysis; companion matrix and companion pencil. SIAM J. Matrix Anal. Appl. 39, 1245–1269 (2018)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Aurentz, J., Mach, T., Robol, L., Vandebril, R., Watkins, D.S.: Fast and backward stable computation of the eigenvalues of matrix polynomials. Math. Comput. 88, 313–347 (2019)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Barnett, S.: A companion matrix analogue for orthogonal polynomials. Linear Algebra Appl. 12, 197–208 (1975)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Barnett, S.: Congenial matrices. Linear Algebra Appl. 41, 277–298 (1981)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Bini, D., Gemignani, L., Pan, V.: Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations. Numer. Math. 100, 373–408 (2005)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Boyd, J.P.: Finding the zeros of a univariate equation: proxy rootfinders, Chebyshev interpolation, and the companion matrix. SIAM Rev. 55, 375–396 (2013)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Brualdi, R., Ryser, H.: Combinatorial Matrix Theory. Cambridge University Press, Cambridge (1991)

    Book  Google Scholar 

  10. 10.

    Bueno, M.I., Curlett, K., Furtado, S.: Structured strong linearizations from Fiedler pencils with repetition I. Linear Algebra Appl. 460, 51–80 (2014)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Bueno, M.I., Curlett, K., Furtado, S.: Structured linearizations from Fiedler pencils with repetition II. Linear Algebra Appl. 463, 282–321 (2014)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Bueno, M.I., De Terán, F.: Eigenvectors and minimal bases for some families of Fiedler-like linearizations. Linear Multilinear Algebra 62, 39–62 (2014)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Bueno, M.I., De Terán, F., Dopico, F.M.: Recovery of eigenvectors and minimal bases of matrix polynomials from generalized Fiedler linearizations. SIAM J. Matrix Anal. Appl. 32, 463–483 (2011)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Bueno, M.I., Dopico, F.M., Pérez, J., Saavedra, R., Zykovsi, B.: A unified approach to Fiedler-like pencils via strong block minimal bases pencils. Linear Algebra Appl. 547, 45–104 (2018)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Carstensen, C.: Linear construction of companion matrices. Linear Algebra Appl. 149, 191–214 (1991)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Corless, R.M.: Generalized companion matrices in the Lagrange basis. In: Gonzalez-Vega, L., Recio, T. (eds.). Proceedings EACA, pp. 317–322 (2004)

  17. 17.

    Corless, R.M.: On a generalized companion matrix pencil for matrix polynomials expressed in the Lagrange basis. In: Wang, D., Zhi, L. (eds.) Symbolic-Numeric Computation. Trends in Mathematics. Birkhäuser, Basel (2007)

    Google Scholar 

  18. 18.

    Day, D., Romero, L.: Roots of polynomials expressed in terms of orthogonal polynomials. SIAM J. Numer. Anal. 43, 1969–1987 (2006)

    MathSciNet  Article  Google Scholar 

  19. 19.

    De Terán, F., Hernando, C.: A class of quasi-sparse companion pencils. In: Bini, D.A. (ed.) Structured Matrices in Numerical Linear Algebra. INdAM series, pp. 157–179. Springer, Berlin (2019)

    Chapter  Google Scholar 

  20. 20.

    De Terán, F., Dopico, F.M., Mackey, D.S.: Fiedler companion linearizations and the recovery of minimal indices. SIAM J. Matrix Anal. Appl. 31, 2181–2204 (2010)

    MathSciNet  Article  Google Scholar 

  21. 21.

    De Terán, F., Dopico, F.M., Mackey, D.S.: Palindromic companion forms for matrix polynomials of odd degree. J. Comput. Appl. Math. 236, 1464–1480 (2011)

    MathSciNet  Article  Google Scholar 

  22. 22.

    De Terán, F., Dopico, F.M., Mackey, D.S.: Fiedler companion linearizations for rectangular matrix polynomials. Linear Algebra Appl. 437, 957–991 (2012)

    MathSciNet  Article  Google Scholar 

  23. 23.

    De Terán, F., Dopico, F.M., Mackey, D.S.: Spectral equivalence of matrix polynomials and the Index Sum Theorem. Linear Algebra Appl. 459, 264–333 (2014)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Deaett, L., Fischer, J., Garnett, C., Vander Meulen, K.N.: Non-sparse companion matrices. Electron. J. Linear Algebra 35, 223–247 (2019)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Dopico, F.M., Lawrence, P., Pérez, J., Van Dooren, P.: Block Kronecker linearizations of matrix polynomials and their backward errors. Numer. Math. 140, 373–426 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Eastman, B., Kim, I.-J., Shader, B.L., Vander Meulen, K.N.: Companion matrix patterns. Linear Algebra Appl. 436, 255–272 (2014)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Eastman, B., Vander Meulen, K.N.: Pentadiagonal companion matrices. Spec. Matrices 4, 13–30 (2016)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Fiedler, M.: A note on companion matrices. Linear Algebra Appl. 372, 325–331 (2003)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Fortune, S.: An iterated eigenvalue algorithm for approximating roots of univariate polynomials. J. Symb. Comput. 33, 627–646 (2002)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Garnett, C., Shader, B.L., Shader, C.L., van den Driessche, P.: Characterization of a family of generalized companion matrices. Linear Algebra Appl. 498, 360–365 (2016)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Gemignani, L.: Structured matrix methods for polynomial root-finding. In: ISSAC07, pp. 175–180, ACM (2007)

  32. 32.

    Gohberg, I., Lancaster, P., Rodman, L.: Matrix Polynomials. Academic Press, New York (1982)

    MATH  Google Scholar 

  33. 33.

    Good, I.J.: The colleague matrix, a Chebyshev analogue of the companion matrix. Q. J. Math Oxford Ser. (2) 12, 61–68 (1961)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Hungerford, T.W.: Algebra. GTM 73. Springer, Berlin (1980)

    Book  Google Scholar 

  35. 35.

    Ma, C., Zhan, X.: Extremal sparsity of the companion matrix of a polynomial. Linear Algebra Appl. 438, 621–625 (2013)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Maroulas, J., Barnett, S.: Polynomials with respect to a general basis I. Theory. J. Math. Anal. Appl. 72, 177–194 (1979)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Nakatsukasa, J., Noferini, V.: On the stability of computing polynomial roots via confederate linearizations. Math. Comp. 85, 2391–2425 (2016)

    MathSciNet  Article  Google Scholar 

  38. 38.

    Prells, U., Friswell, M.I., Garvey, S.D.: Use of geometric algebra: compound matrices and the determinant of the sum of two matrices. Proc. R. Soc. Lond. A 453, 273–285 (2003)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Van Dooren, P., Dewilde, P.: The eigenstructure of an arbitrary polynomial matrix: computational aspects. Linear Algebra Appl. 50, 545–579 (1983)

    MathSciNet  Article  Google Scholar 

  40. 40.

    Vologiannidis, S., Antoniou, E.N.: A permuted factors approach for the linearization of polynomial matrices. Math. Control Signals Syst. 22, 317–342 (2011)

    MathSciNet  Article  Google Scholar 

  41. 41.

    Werner, W.: A generalized companion matrix of a polynomial and some applications. Linear Algebra Appl. 55, 19–36 (1983)

    MathSciNet  Article  Google Scholar 

Download references


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].

Author information



Corresponding author

Correspondence to Fernando De Terán.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

De Terán, F., Hernando, C. A note on generalized companion pencils in the monomial basis. RACSAM 114, 8 (2020).

Download citation


  • 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