A Note on Classes of Structured Matrices with Elliptical Type Numerical Range

Abstract

We identify new classes of structured matrices whose numerical range is of the elliptical type, that is, an elliptical disk or the convex hull of elliptical disks.

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

References

  1. [1]

    N. Bebiano, J. da Providência, A. Nata: The numerical range of banded biperiodic Toeplitz operators. J. Math. Anal. Appl. 398 (2013), 189–197.

    MathSciNet  Article  Google Scholar 

  2. [2]

    N. Bebiano, S. Furtado: Remarks on anti-tridiagonal matrices. Appl. Math. Comput. 373 (2020), Article ID 125008, 10 pages.

  3. [3]

    N. Bebiano, I. M. Spitkovsky: Numerical ranges of Toeplitz operators with matrix symbols. Linear Algebra Appl. 436 (2012), 1721–1726.

    MathSciNet  Article  Google Scholar 

  4. [4]

    E. S. Brown, I. M. Spitkovsky: On matrices with elliptical numerical ranges. Linear Multilinear Algebra 52 (2004), 177–193.

    MathSciNet  Article  Google Scholar 

  5. [5]

    M.-T. Chien: On the numerical range of tridiagonal operators. Linear Algebra Appl. 246 (1996), 203–214.

    MathSciNet  Article  Google Scholar 

  6. [6]

    M.-T. Chien, H. Nakazato: The numerical range of a tridiagonal operator. J. Math. Anal. Appl. 373 (2011), 297–304.

    MathSciNet  Article  Google Scholar 

  7. [7]

    R. T. Chien, I. M. Spitkovsky: On the numerical ranges of some tridiagonal matrices. Linear Algebra Appl. 470 (2015), 228–240.

    MathSciNet  Article  Google Scholar 

  8. [8]

    M.-T. Chien, L. Yeh, Y.-T. Yeh, F.-Z. Lin: On geometric properties of the numerical range. Linear Algebra Appl. 274 (1998), 389–410.

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. Eiermann: Field of values and iterative methods. Linear Algebra Appl. 180 (1993), 167–197.

    MathSciNet  Article  Google Scholar 

  10. [10]

    T. Geryba, I. M. Spitkovsky: On the numerical range of some block matrices with scalar diagonal blocks. To appear in Linear and Multilinear Algebra.

  11. [11]

    K. E. Gustafson, D. K. M. Rao: Numerical Range: The Field of Values of Linear Operators and Matrices. Universitext. Springer, New York, 1997.

    Google Scholar 

  12. [12]

    R. A. Horn, C. R. Johnson: Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991.

    Google Scholar 

  13. [13]

    R. Kippenhahn: Über den Wertevorrat einer Matrix. Math. Nachr. 6 (1951), 193–228. (In German.)

    MathSciNet  Article  Google Scholar 

  14. [14]

    E. M. Klein: The numerical range of a Toeplitz operator. Proc. Am. Math. Soc. 35 (1972), 101–103.

    MathSciNet  Article  Google Scholar 

  15. [15]

    M. Marcus, C. Pesce: Computer generated numerical ranges and some resulting theorems. Linear and Multilinear Algebra 20 (1987), 121–157.

    MathSciNet  Article  Google Scholar 

  16. [16]

    P. X. Rault, T. Sendova, I. M. Spitkovsky: 3-by-3 matrices with elliptical numerical range revisited. Electron. J. Linear Algebra 26 (2013), 158–167.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Susana Furtado.

Additional information

This work was partially supported by project UID/MAT/00324/2019 and by project UID/MAT/04721/2020.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Bebiano, N., Furtado, S. A Note on Classes of Structured Matrices with Elliptical Type Numerical Range. Czech Math J (2021). https://doi.org/10.21136/CMJ.2021.0174-20

Download citation

Keywords

  • tridiagonal matrix
  • antitridiagonal matrix
  • elliptical disk
  • numerical range

MSC 2020

  • 15A21
  • 15A60