Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 5 (2018), 803-826.

Symmetric numerical ranges of four-by-four matrices

Shelby L. Burnett, Ashley Chandler, and Linda J. Patton

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Numerical ranges of matrices with rotational symmetry are studied. Some cases in which symmetry of the numerical range implies symmetry of the spectrum are described. A parametrized class of 4×4 matrices K(a) such that the numerical range W(K(a)) has fourfold symmetry about the origin but the generalized numerical range WK(a)(K(a)) does not have this symmetry is included. In 2011, Tsai and Wu showed that the numerical ranges of weighted shift matrices, which have rotational symmetry about the origin, are also symmetric about certain axes. We show that any 4×4 matrix whose numerical range has fourfold symmetry about the origin also has the corresponding axis symmetry. The support function used to prove these results is also used to show that the numerical range of a composition operator on Hardy space with automorphic symbol and minimal polynomial z41 is not a disk.

Article information

Involve, Volume 11, Number 5 (2018), 803-826.

Received: 13 December 2016
Revised: 30 July 2017
Accepted: 3 September 2017
First available in Project Euclid: 12 April 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05]
Secondary: 47B33: Composition operators

numerical range symmetry weighted shift matrices composition operator


Burnett, Shelby L.; Chandler, Ashley; Patton, Linda J. Symmetric numerical ranges of four-by-four matrices. Involve 11 (2018), no. 5, 803--826. doi:10.2140/involve.2018.11.803.

Export citation


  • N. Bebiano and I. M. Spitkovsky, “Numerical ranges of Toeplitz operators with matrix symbols”, Linear Algebra Appl. 436:6 (2012), 1721–1726.
  • P. S. Bourdon and J. H. Shapiro, “The numerical ranges of automorphic composition operators”, J. Math. Anal. Appl. 251:2 (2000), 839–854.
  • W.-S. Cheung and N.-K. Tsing, “The $C$-numerical range of matrices is star-shaped”, Linear and Multilinear Algebra 41:3 (1996), 245–250.
  • M.-T. Chien and H. Nakazato, “Singular points of the ternary polynomials associated with 4-by-4 matrices”, Electron. J. Linear Algebra 23 (2012), 755–769.
  • C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic functions, CRC Press, Boca Raton, FL, 1995.
  • L. Deaett, R. H. Lafuente-Rodriguez, J. Marin, Jr., E. Haller Martin, L. J. Patton, K. Rasmussen, and R. B. Johnson Yates, “Trace conditions for symmetry of the numerical range”, Electron. J. Linear Algebra 26 (2013), 591–603.
  • G. Fischer, Plane algebraic curves, Student Mathematical Library 15, American Mathematical Society, Providence, RI, 2001.
  • H.-L. Gau and P. Y. Wu, “Line segments and elliptic arcs on the boundary of a numerical range”, Linear Multilinear Algebra 56:1-2 (2008), 131–142.
  • C. G. Gibson, Elementary geometry of algebraic curves: an undergraduate introduction, Cambridge University Press, 1998.
  • M. Goldberg and E. G. Straus, “Elementary inclusion relations for generalized numerical ranges”, Linear Algebra and Appl. 18:1 (1977), 1–24.
  • K. E. Gustafson and D. K. M. Rao, Numerical range: the field of values of linear operators and matrices, Springer, 1997.
  • T. R. Harris, M. Mazzella, L. J. Patton, D. Renfrew, and I. M. Spitkovsky, “Numerical ranges of cube roots of the identity”, Linear Algebra Appl. 435:11 (2011), 2639–2657.
  • F. Hausdorff, “Der Wertvorrat einer Bilinearform”, Math. Z. 3:1 (1919), 314–316.
  • M. T. Heydari and A. Abdollahi, “The numerical range of finite order elliptic automorphism composition operators”, Linear Algebra Appl. 483 (2015), 128–138.
  • R. A. Horn and C. R. Johnson, Topics in matrix analysis, Cambridge University Press, Cambridge, 1991.
  • D. S. Keeler, L. Rodman, and I. M. Spitkovsky, “The numerical range of $3\times 3$ matrices”, Linear Algebra Appl. 252 (1997), 115–139.
  • R. Kippenhahn, “Über den Wertevorrat einer Matrix”, Math. Nachr. 6 (1951), 193–228.
  • R. Kippenhahn, “On the numerical range of a matrix”, Linear Multilinear Algebra 56:1-2 (2008), 185–225.
  • K. Lentzos and L. Pasley, “Determinantal representations of invariant hyperbolic plane curves”, preprint, 2017.
  • C.-K. Li, “$C$-numerical ranges and $C$-numerical radii”, Linear and Multilinear Algebra 37:1-3 (1994), 51–82.
  • C.-K. Li and N.-K. Tsing, “Matrices with circular symmetry on their unitary orbits and $C$-numerical ranges”, Proc. Amer. Math. Soc. 111:1 (1991), 19–28.
  • L. J. Patton, “Some block Toeplitz composition operators”, J. Math. Anal. Appl. 400:2 (2013), 363–376.
  • L. Rodman and I. M. Spitkovsky, “$3\times3$ matrices with a flat portion on the boundary of the numerical range”, Linear Algebra Appl. 397 (2005), 193–207.
  • B.-S. Tam and S. Yang, “On matrices whose numerical ranges have circular or weak circular symmetry”, Linear Algebra Appl. 302/303 (1999), 193–221.
  • O. Toeplitz, “Das algebraische Analogon zu einem Satze von Fejér”, Math. Z. 2:1-2 (1918), 187–197.
  • M. C. Tsai and P. Y. Wu, “Numerical ranges of weighted shift matrices”, Linear Algebra Appl. 435:2 (2011), 243–254.
  • S.-H. Tso and P. Y. Wu, “Matricial ranges of quadratic operators”, Rocky Mountain J. Math. 29:3 (1999), 1139–1152.
  • F. A. Valentine, Convex sets, McGraw-Hill, 1964.
  • R. Westwick, “A theorem on numerical range”, Linear and Multilinear Algebra 2 (1975), 311–315.
  • P. Y. Wu, “Numerical ranges as circular discs”, Appl. Math. Lett. 24:12 (2011), 2115–2117.