Annals of Functional Analysis

Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bounds for the zeros of polynomials

Amer Abu-Omar and Fuad Kittaneh

Full-text: Open access


We apply numerical radius and spectral radius estimates to the Frobenius companion matrices of monic polynomials to derive new bounds for their zeros and give different proofs of some known bounds.

Article information

Ann. Funct. Anal., Volume 5, Number 1 (2014), 56-62.

First available in Project Euclid: 5 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47A12: Numerical range, numerical radius
Secondary: 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory [See also 65F35, 65J05] 26C10: Polynomials: location of zeros [See also 12D10, 30C15, 65H05] 30C15: Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}

Frobenius companion matrix zeros of polynomials numerical radius spectral radius inequality


Abu-Omar, Amer; Kittaneh, Fuad. Estimates for the numerical radius and the spectral radius of the Frobenius companion matrix and bounds for the zeros of polynomials. Ann. Funct. Anal. 5 (2014), no. 1, 56--62. doi:10.15352/afa/1391614569.

Export citation


  • A.A. Abdurakhmanov, Geometry of a Hausdorff domain in problems of localization of the spectrum of arbitrary matrices, Math. USSR Sb. 59 (1988), 39–51.
  • A. Abu-Omar and F. Kittaneh, A numerical radius inequality involving the generalized Aluthge transform, Studia Math. 216 (2013), 69–75.
  • A. Abu-Omar and F. Kittaneh, Numerical radius inequalities for $ n\times n$ operator matrices, Linear Algebra Appl. (to appear).
  • M.L. Buzano, Generalizzatione della diseguaglianza di Cauchy-Schwarz (Italian), Rend. Sem. Mat. Univ. e Politech. Torino 31 (1971/73), 405–409 (1974).
  • M. Fujii and F. Kubo, Buzano's inequality and bounds for roots of algebraic equations, Proc. Amer. Math. Soc. 117 (1993), 359–361.
  • K.E. Gustafson and D.K.M. Rao, Numerical Range, Springer, New York, 1997.
  • R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, 1985.
  • J.C. Hou and H.K. Du, Norm inequalities of positive operator matrices, Integral Equations Operator Theory 22 (1995), 281–294.
  • F. Kittaneh, Bounds for the zeros of polynomials from matrix inequalities, Arch. Math. (Basel) 81 (2003), 601–608.
  • F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 (2003), 11–17.
  • F. Kittaneh and K. Shebrawi, Bounds and majorization relations for the zeros of polynomials, Numer. Funct. Anal. Optim. 30 (2009), 98–110.
  • K. Paul and S. Bag, On numerical radius of a matrix and estimation of bounds for zeros of a polynomial, Int. J. Math. Math. Sci. (2012), Art. ID 129132, 15 pp.
  • T. Yamazaki, On upper and lower bounds of the numerical radius and an equality condition, Studia Math. 178 (2007), 83–89.