Kyungpook Mathematical Journal 2021; 61(3): 583-590
Published online September 30, 2021
Copyright © Kyungpook Mathematical Journal.
New Bounds for the Numerical Radius of a Matrix in Terms of Its Entries
Department of Mathematics, Mustapha Stambouli University, Mascara, Algeria
e-mail : firstname.lastname@example.org
Received: March 18, 2020; Revised: January 15, 2021; Accepted: February 8, 2021
In this work we give new upper and lower bounds for the numerical radius of a complex square matrix A using the entries and the trace of A.
Keywords: Numerical range, trace of matrix, Frobenius norm, numerical radius, spectral radius.
The numericalrange of a complex
The numerical radius
Numerous contributions related to numerical radius were made by various people including M. Goldberg, E. Tadmor and G. Zwas , also J. Merikoski and R. Kumar . We cite here some properties of the numerical radius which are well known see . Let
In this paper, without knowing thenumerical radius of the matrix
It is well known, see , that
In  C. R. Johnson gave an upper bound for the numerical radius
J. K. Merikoski and R. Kumar  gave some lower bounds for the numerical radius
2. Bounds For the Numerical Radius
In this section, we give some upper and lowers bounds for the numerical radius of a given complex
Proposition 2.1. For any matrix
Theorem 2.2. Let
Corollary 2.3. Let
Theorem 2.4. Let
We have used the fact that
Corollary 2.5. Let
Theorem 2.6. Let
Theorem 2.7. Let
Theorem 2.8. Let
Lemma 2.9. If
Corollary 2.10. Let
Theorem 2.11. Let
Theorem 2.12. Let
Using the previous theorem then the required statement follows immediately.
Theorem 2.13. Let
It is clear that the choice
Corollary 2.14. Let
Theorem 2.15. Let
3. The Areal Numerical Radius of Matrices
R. A. Smith and L. Mirsky in  called areal spread of the matrix
Theorem 3.1. Let
where the supremum is taken over all nonzero
The author would like to thank the reviewers for their very helpful comments and suggestions.
- M. Goldberg, E. Tadmor and G. Zwas, The numerical radius and spectral matrices, Linear and Multilinear Algebra, 2(1975), 317-326.
- R. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
- C. R. Johnson, A gersgorin inclusion set for the field of values of a finite matrix, Proceedings of the american mathematical society. 41(1973), 57-60.
- J. K. Merikoski and R. Kumar, Lower bounds for the numerical radius, Linear Algebra Appl., 410(2005), 135-142.
- R. A. Smith and L. Mirsky, The areal spread of matrices, Linear Algebra Appl., 2(1969), 127-129.