Skip to main content

An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues

Abstract

The construction of totally nonnegative (TN) matrices with prescribed eigenvalues is an important topic in real-valued nonnegative inverse eigenvalue problems. TN matrices are square matrices whose minors are all nonnegative. Our previous paper (Numer. Algor. 70, 469–484, ??2015) presented a finite-step construction of TN matrices limited to upper or lower Hessenberg forms with prescribed eigenvalues, based on the discrete hungry Toda (dhToda) equation which is derived from the study of integrable systems. Building on our previous paper, we produce the construction of banded TN matrices with an arbitrary number of diagonals in both lower and upper triangular parts and prescribed eigenvalues, involving upper Hessenberg, lower Hessenberg, and dense TN matrices with prescribed eigenvalues. We first prepare an infinite sequence associated with distinct eigenvalues \(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\) and two integers M and N which determine the upper and lower bandwidths of m-by-m banded matrices, respectively. Both M and N play a key role for achieving our purpose. The study follows similar lines to our previous paper, but is complicated by the introduction of N. We next consider extended Hankel determinants and extended Hadamard polynomials involving elements of the infinite sequence and then derive their relationships. These relationships help us understand banded TN matrices with eigenvalues \(\lambda _{1},\lambda _{2},\dots ,\lambda _{m}\) from the viewpoint of an extension of the dhToda equation. Finally, we propose a finite-step procedure for constructing TN matrices with an arbitrary upper and lower bandwidths and prescribed eigenvalues and also give illustrative examples.

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

References

  1. Akaiwa, K., Nakamura, Y., Iwasaki, M., Tsutsumi, H., Kondo, K.: A finite-step construction of totally nonnegative matrices with specified eigenvalues. Numer. Algor. 70, 469–484 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  2. Ando, T.: Totally positive matrices. Linear Algebra Appl. 90, 165–219 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  3. Chu, M.T., Golub, G.H.: Inverse Eigenvalue Problems: Theory, Algorithms, and Applications. Oxford Univ. Press, New York (2005)

    Book  MATH  Google Scholar 

  4. Fallat, S.: Bidiagonal factorizations of totally nonnegative matrices. Amer. Math. Monthly 108, 697–712 (2001)

    MathSciNet  Article  MATH  Google Scholar 

  5. Fallat, S., Gekhtman, M.I.: Jordan structures of totally nonnegative matrices. Canad. J. Math. 57, 82–98 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  6. Fallat, S.M., Johnson, C.R.: Totally Nonnegative Matrices. Princeton Univ. Press, Princeton (2011)

    Book  MATH  Google Scholar 

  7. Fukuda, A., Ishiwata, E., Yamamoto, Y., Iwasaki, M., Nakamura, Y.: Integrable discrete hungry systems and their related matrix eigenvalues. Annal. Mat. Pura Appl. 192, 423–445 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  8. Gasca, M., Micchelli, C.A. (eds.): Total Positivity and Its Applications. Kluwer Academic, Dordrecht (1996)

  9. Gladwell, G.M.L.: Inner totally positive matrices. Linear Algebra Appl. 393, 179–195 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  10. The GNU Multiple Precision Arithmetic Library, https://gmplib.org

  11. The GNU MPFR Library, http://www.mpfr.org

  12. Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins Univ. Press, Baltimore (2013)

  13. Henrici, P.: Applied and Computational Complex Analysis, vol. 1. John Wiley, New York (1974)

    MATH  Google Scholar 

  14. Higham, N.J.: Accuracy and Stability of Numerical Algorithms, 2nd edn. SIAM, Philadelphia (2002)

  15. Hirota, R.: The direct method in soliton theory. Cambridge Univ. press (2004)

  16. In: Hogben, L. (ed.) : Handbook of Linear Algebra, 2nd edn. CRC Press (2014)

  17. Iwasaki, M., Nakamura, Y.: On the convergence of a solution of the discrete Lotka-Volterra system. Inverse Probl. 18, 1569–1578 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  18. Karlin, S.: Total Positivity, vol. 1. Stanford Univ. Press, Redwood city (1968)

    MATH  Google Scholar 

  19. Matrix Market, http://math.nist.gov/MatrixMarket/

  20. Pinkus, A.: Totally Positive Matrices. Cambridge Univ. Press, New York (2009)

    Book  MATH  Google Scholar 

  21. Press, W.H., Vetterling, W.T., Teukolsky, S.A., Flannery, B.P.: Numerical Recipes in C, 2nd edn. Cambridge Univ. Press, Cambridge (1992)

  22. Rutishauser, H.: Bestimmung der Eigenwerte und Eigenvektoren einer Matrix mit Hilfe des Quotienten-Differenzen-Algorithmus. Z. Angew. Math. Phys. 6, 387–401 (1955)

    MathSciNet  Article  MATH  Google Scholar 

  23. Yamamoto, Y., Fukaya, T.: Differential qd algorithm for totally nonnegative band matrices: convergence properties and error analysis. JSIAM Letters 1, 56–59 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  24. Tokihiro, T., Nagai, A., Satsuma, J.: Proof of solitonical nature of box and ball system by the means of inverse ultra-discretization. Inverse Probl. 15, 1639–1662 (1999)

    MathSciNet  Article  MATH  Google Scholar 

  25. Zhang, W., Higham, N.J.: Matrix Depot: An extensible test matrix collection for Julia. Peer J Comput. Sci. 2:e58 (2016)

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Kanae Akaiwa.

Additional information

The authors thank the reviewers for their careful reading and insightful suggestions. This work is supposed by the Grant-in-Aid for JSPS Fellows No. 26 ⋅6045 and the Grant-in-Aid for Scientific Research (C) No. 26400208 from the Japan Society for the Promotion of Science.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Akaiwa, K., Nakamura, Y., Iwasaki, M. et al. An arbitrary band structure construction of totally nonnegative matrices with prescribed eigenvalues. Numer Algor 75, 1079–1101 (2017). https://doi.org/10.1007/s11075-016-0231-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11075-016-0231-7

Keywords

  • Extended discrete hungry Toda equation
  • Finite-step construction
  • Inverse eigenvalue problem
  • Totally nonnegative

Mathematics Subject Classifications (2010)

  • 65F18
  • 15A48
  • 37N30