Some New Localization Results for Eigenvalues of Complex Matrices

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DOI: https://doi.org/10.28924/2291-8639-24-2026-222
Published: Aug 3, 2026
Cite as:
M. Rehman, Riffat Amna, Saima Akram, T.H. Rasulov, N.H. Mamatova, J.Sh. Ostonov, B.Z. Gafurov, M.H. Boboqulova, Some New Localization Results for Eigenvalues of Complex Matrices, Int. J. Anal. Appl., 24 (2026), 222.

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M. Rehman, Riffat Amna, Saima Akram, T.H. Rasulov, N.H. Mamatova, J.Sh. Ostonov, B.Z. Gafurov, M.H. Boboqulova

Abstract

The localization of spectrum of complex-valued matrices is a foundational tool in matrix analysis and applied mathematics which offers a concise and computable descriptions on the location of spectrum in the complex plane. Many effective localization bounds, for instance, Gorgerin-type regions, and pseudo-spectrum allows to study and analyze the stability, sensitivity to perturbations, and spectral separation without full eigenvalue computation. In this paper, we present some new results on the localization of the spectrum of complex-valued matrices. Our approach is based on collection of various tools from matrix theory and numerical linear algebra. We have used MATLAB to demonstrate the localization of spectrum contained in the Gershgorin disks in the complex plane.

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References

  1. A. Dall'Acqua, D. Mugnolo, M. Schelling, A New Gershgorin-Type Result for the Localisation of the Spectrum of Matrices, arXiv:1409.5133, 2014. https://doi.org/10.48550/arXiv.1409.5133.
  2. G.V. Demidenko, V.S. Prokhorov, On Location of the Matrix Spectrum with Respect to a Parabola, Siberian Adv. Math. 33 (2023), 190-199. https://doi.org/10.1134/s1055134423030033.
  3. G.V. Demidenko, Z. Wang, Localization of the Matrix Spectrum and Lyapunov Type Equations, arXiv:2312.12177, 2023. https://doi.org/10.48550/arXiv.2312.12177.
  4. S.A. Gershgorin, Über die Abgrenzung der Eigenwerte einer Matrix, Izv. Akad. Nauk, Ser. Mat. 7 (1931), 749-754.
  5. L. Kolotilina, Generalizations of the Ostrowski–Brauer Theorem, Linear Algebra Appl. 364 (2003), 65-80. https://doi.org/10.1016/s0024-3795(02)00537-2.
  6. R. Marsli, F.J. Hall, Further Results on Geršgorin Discs, Linear Algebra Appl. 439 (2013), 189-195. https://doi.org/10.1016/j.laa.2013.02.021.
  7. R. Marsli, F.J. Hall, Geometric Multiplicities and Geršgorin Discs, Am. Math. Mon. 120 (2013), 452-455. https://doi.org/10.4169/amer.math.monthly.120.05.452.
  8. R. Marsli, F.J. Hall, Some Refinements of Gersgorin Discs, Int. J. Algebra 7 (2013), 573-580. https://doi.org/10.12988/ija.2013.3763.
  9. R. Marsli, F. Hall, On the Location of Eigenvalues of Real Matrices, Electron. J. Linear Algebra 32 (2017), 357-364. https://doi.org/10.13001/1081-3810.3544.
  10. J.M. Peña, Eigenvalue Localization for Symmetric Positive Toeplitz Matrices, Axioms 14 (2025), 232. https://doi.org/10.3390/axioms14040232.
  11. B. Solmaz, T. Ayyildiz, Certified Real Eigenvalue Location, arXiv:2601.14491, 2026. https://doi.org/10.48550/arXiv.2601.14491.
  12. R.S. Varga, Gershgorin and His Circles, Springer, 2011.
  13. J. Wu, D. Wang, J. Zhao, Localization of Eigenvalues and Estimation of the Spread for Complex Matrices, Izv. Math. 79 (2015), 208-215. https://doi.org/10.1070/IM2015v079n01ABEH002739.
  14. J. Wu, J. Zhao, A Survey of the Progress of Locating Methods of Complex Matrices’ Eigenvalues and Some New Location Theorem and Their Applications, IMA J. Appl. Math. 80 (2014), 1273-1285. https://doi.org/10.1093/imamat/hxu043.