Some new estimations on the spectral radius of the Schur product of matrices

Fubin Chen; Fubin Chen

doi:10.3934/math.2025006

AIMS Mathematics

2025, Volume 10, Issue 1: 97-116. doi: 10.3934/math.2025006

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Research article

Some new estimations on the spectral radius of the Schur product of matrices

Fubin Chen ^,

Department of Engineering, Oxbridge College of Kunming University of Science and Technology, Kunming 650106, China

Received: 15 August 2024 Revised: 12 December 2024 Accepted: 18 December 2024 Published: 02 January 2025
MSC : 15A15, 15A42

The present study investigates the Schur product of multiple nonnegative matrices $ {{\boldsymbol{G}}_1} $, $ {{\boldsymbol{G}}_2} $, $ \cdots $, $ {{\boldsymbol{G}}_n} $. By utilizing the Perron root estimation for nonnegative matrices and applying the Hölder inequality, we establish some upper bounds on $ \rho \left( {{{\boldsymbol{G}}_1} \circ {{\boldsymbol{G}}_2} \circ \cdots \circ {{\boldsymbol{G}}_n}} \right) $. These novel findings encompass and extend certain earlier results. Some comparative analysis between our new results and existing results is conducted through numerical experiments. Theoretical analysis and data calculations demonstrate that our results outperform those reported in other studies.
- nonnegative matrix,
- Schur product,
- spectral radius,
- upper bound,
- irreducible
Citation: Fubin Chen. Some new estimations on the spectral radius of the Schur product of matrices[J]. AIMS Mathematics, 2025, 10(1): 97-116. doi: 10.3934/math.2025006

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Abstract

The present study investigates the Schur product of multiple nonnegative matrices $ {{\boldsymbol{G}}_1} $, $ {{\boldsymbol{G}}_2} $, $ \cdots $, $ {{\boldsymbol{G}}_n} $. By utilizing the Perron root estimation for nonnegative matrices and applying the Hölder inequality, we establish some upper bounds on $ \rho \left( {{{\boldsymbol{G}}_1} \circ {{\boldsymbol{G}}_2} \circ \cdots \circ {{\boldsymbol{G}}_n}} \right) $. These novel findings encompass and extend certain earlier results. Some comparative analysis between our new results and existing results is conducted through numerical experiments. Theoretical analysis and data calculations demonstrate that our results outperform those reported in other studies.

References

[1]	R. A. Horn, C. R. Johnson, Topics in matrix analysis, Cambridge: Cambridge University Press, 1991. https://doi.org/10.1017/CBO9780511840371
[2]	M. Z. Fang, Bounds on eigenvalues of the Hadamard product and the Fan product of matrices, Linear Algebra Appl., 425 (2007), 7–15. https://doi.org/10.1016/j.laa.2007.03.024 doi: 10.1016/j.laa.2007.03.024
[3]	R. Huang, Some inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl., 428 (2008), 1551–1559. https://doi.org/10.1016/j.laa.2007.10.001 doi: 10.1016/j.laa.2007.10.001
[4]	Q. B. Liu, G. L. Chen, On two inequalities for the Hadamard product and the Fan product of matrices, Linear Algebra Appl., 431 (2009), 974–984. https://doi.org/10.1016/j.laa.2009.03.049 doi: 10.1016/j.laa.2009.03.049
[5]	Q. B. Liu, G. L. Chen, L. L. Zhao, Some new bounds on the spectral radius of matrices, Linear Algebra Appl., 432 (2010), 936–948. https://doi.org/10.1016/j.laa.2009.10.006 doi: 10.1016/j.laa.2009.10.006
[6]	Q. P. Guo, J. S. Leng, H. B. Li, C. Cattani, Some bounds on eigenvalues of the Hadamard product and the Fan product of matrices, Mathematics, 7 (2019), 147. https://doi.org/10.3390/math7020147 doi: 10.3390/math7020147
[7]	Q. Zhong, Some new inequalities for nonnegative matrices involving Schur product, AIMS Mathematics., 8 (2023), 29667–29680. https://doi.org/10.3934/math.20231518 doi: 10.3934/math.20231518
[8]	J. Li, H. Hai, Some new inequalities for the Hadamard product of nonnegative matrices, Linear Algebra Appl., 606 (2020), 159–169. https://doi.org/10.1016/j.laa.2020.07.025 doi: 10.1016/j.laa.2020.07.025
[9]	K. M. R. Audenaert, Spectral radius of Hadamard product versus conventional product for nonnegative matrices, Linear Algebra Appl., 432 (2010), 366–368. https://doi.org/10.1016/j.laa.2009.08.017 doi: 10.1016/j.laa.2009.08.017
[10]	Z. J. Huang, On the spectral radius and the spectral norm of Hadamard products of nonnegative matrices, Linear Algebra Appl., 434 (2011), 457–462. https://doi.org/10.1016/j.laa.2010.08.038 doi: 10.1016/j.laa.2010.08.038
[11]	L. L. Zhao, Q. B. Liu, Some inequalities on the spectral radius of matrices, J. Inequal. Appl., 2018 (2018), 5. https://doi.org/10.1186/s13660-017-1598-2 doi: 10.1186/s13660-017-1598-2
[12]	R. A. Horn, C. R. Johnson, Matrix analysis, Cambridge: Cambridge University Press, 1985. https://doi.org/10.1017/CBO9781139020411
[13]	A. Brauer, Limits for the characteristic roots of a matrix. Ⅱ, Duke Math. J., 14 (1947), 21–26. https://doi.org/10.1215/s0012-7094-47-01403-8 doi: 10.1215/s0012-7094-47-01403-8
[14]	G. H. Hardy, J. E. Littlewood, G. Pólya, Inequalities, Cambridge: Cambridge University Press, 1952.

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