A Note on Gershgorin Disks in the Elliptic Plane

Arzu Sürekçi; Hidayet Hüda Kösal; Mehmet Ali Güngör

doi:10.33187/jmsm.986344

EN

A Note on Gershgorin Disks in the Elliptic Plane

Abstract

In this study, we derive Gershgorin discs of elliptic complex matrices in the elliptic plane. Also, we investigate the location of the zeros of an elliptic complex valued polynomial with the help of Gershgorin discs of elliptic complex matrices. To prove the authenticity of our results and to distinguish them from existing ones, some illustrative examples are also given. Elliptic complex numbers are a generalized form of complex and so real numbers. Thus, the obtained results extend, generalize and complement some known Gershgorin discs results from the literature.

Keywords

Elliptic complex matrix, Elliptic complex numbers, Elliptic valued polynomial, Gershgorin disks

References

[1] F.A. Aliev, V.B. Larin, Optimization of linear control systems, USA, CRC Press, 1998.
[2] D. Calvetti and L. Reichel, Application of ADI iterative methods to the restoration of noisy images, SIAM J. Matrix Anal. Appl., 17(1) (1996), 165-186.
[3] L. Dieci, M.R. Osborne and R.D.Russel, A Riccati transformation method for solving linear BVPs. I: Theoretical Aspects, SIAM J. Numer. Anal., 25(5) (1988), 1055-1073.
[4] W.H. Enright, Improving the efficiency of matrix operations in the numerical solution of stiff ordinary differential equations, ACM Trans. Math. Software, 4(2) (1978), 127-136.
[5] M.A. Epton, Methods for the solution of AXD - BXC = E and its applications in the numerical solution of implicit ordinary differential equations, BIT Numer. Math., 20 (1980) 341-345.
[6] F.R. Gantmacher, The theory of matrices, New York, Chelsea Publishing Company, 1959.
[7] A. Jameson, Solution of the equation ax+xb = c by inversion of an mm or nn matrix, SIAM J. Appl. Math., 16(5) (1968), 1020-1023.
[8] E. Souza, Controllability, observability and the solution of ax􀀀xb = c, Linear Algebra Appl., 39 (1981), 167-188.
[9] M. Dehghan and M. Hajarian, Efficient iterative method for solving the second-order Sylvester matrix equation EVF2 􀀀AVF 􀀀CV = BW, IET Control Theory Appl., 3(10) (2009), 1401-1408.
[10] A. Wu, E. Zhang and F. Liu, On closed-form solutions to the generalized Sylvester-conjugate matrix equation, Appl. Math. Comput., 218(19) (2012), 9730-9741.

[11] I.M. Yaglom, Complex numbers in geometry, New York and London, Academic Press, 1968.
[12] I.M. Yaglom, A simple non-Euclidean geometry and its physical basis, New York, Springer Verlag, 1979.
[13] N. G¨urses, M. Akbiyik and S. Y¨uce, One-parameter homothetic motions and Euler-Savary formula in generalized complex number plane CJ , Adv. Appl. Clifford Algebr., 26(1) (2016), 115-136.
[14] F.S. D¨undar, S. Ersoy and N.T.S. Pereira, Bobillier formula for the elliptical harmonic motion, An. St. Univ. Ovidius Constanta, 26(1) (2018), 103-110.
[15] V. Brodsky and M. Shoham, Dual numbers representation of rigid body dynamics, Mech. Mach. Theory, 34(5) (1999), 693-718.
[16] H.H. Cheng, Programming with dual numbers and its applications in mechanisms design, Eng. Comput., 10 (1994) 212-229.
[17] S. Ulrych, Relativistic quantum physics with hyperbolic numbers, Phys. Lett. B, 625(3-4) (2005), 313-323.
[18] M. Kobayashi, Hyperbolic hopfield neural networks, IEEE Trans. Neural Netw. Learn. Syst., 24 (2013) 335-341.
[19] K. O¨ zen and M. Tosun, p-Trigonometric approach to elliptic biquaternions, Adv. Appl. Clifford Algebr., 28(62) (2018).
[20] J.H. Silverman, The arithmetic of elliptic curve, Graduate Texts in Mathematics, New York, 1988.
[21] S. Basu and D.J. Velleman, On Gauss’s first proof of the fundamental theorem of algebra, Amer. Math. Monthly, 124(8) (2017), 688-694.
[22] H.H. K¨osal, On the Commutative quaternion matrices, Ph. D. Thesis, Sakarya University, 2016.
[23] A. Melman, Modified Gershgorin discs for companion matrices, Soc. Ind. Appl. Math., 54(2) (2012), 355-373.
[24] Y.A. Alpin, M.T. Chien and L. Yeh, The numerical radius and bounds for zeros of a polynomial, Proc. Amer. Math. Soc., 131(3) (2003), 725-730.
[25] H.E. Bell, Gershgorin’s theorem and the zeros of polynomials, Amer. Math. Monthly, 72(3) (1965), 292-295.
[26] A. Edelman and H. Murakami, Polynomial roots from companion matrix eigenvalues, Math. Comp., 64(210) (1995), 763-776.
[27] H. Linden, Bounds for the zeros of polynomials from eigenvalues and singular values of some companion matrices, Linear Algebra Appl., 271(1-3) (1998), 41-82.
[28] H.S Wilf, Perron-Frobenius theory and the zeros of polynomials, Proc. Amer. Math. Soc., 12 (1961), 247-250.

Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Arzu Sürekçi ^*
0000-0003-2003-3507
Türkiye

Hidayet Hüda Kösal
0000-0002-4083-462X
Türkiye

Mehmet Ali Güngör
0000-0003-1863-3183
Türkiye

Publication Date

December 27, 2021

Submission Date

August 23, 2021

Acceptance Date

October 16, 2021

Published in Issue

Year 2021 Volume: 4 Number: 3

DOI

https://doi.org/10.33187/jmsm.986344

IZ

https://izlik.org/JA34DS73RD

APA

Sürekçi, A., Kösal, H. H., & Güngör, M. A. (2021). A Note on Gershgorin Disks in the Elliptic Plane. Journal of Mathematical Sciences and Modelling, 4(3), 104-109. https://doi.org/10.33187/jmsm.986344

AMA

1.Sürekçi A, Kösal HH, Güngör MA. A Note on Gershgorin Disks in the Elliptic Plane. Journal of Mathematical Sciences and Modelling. 2021;4(3):104-109. doi:10.33187/jmsm.986344

Chicago

Sürekçi, Arzu, Hidayet Hüda Kösal, and Mehmet Ali Güngör. 2021. “A Note on Gershgorin Disks in the Elliptic Plane”. Journal of Mathematical Sciences and Modelling 4 (3): 104-9. https://doi.org/10.33187/jmsm.986344.

EndNote

Sürekçi A, Kösal HH, Güngör MA (December 1, 2021) A Note on Gershgorin Disks in the Elliptic Plane. Journal of Mathematical Sciences and Modelling 4 3 104–109.

IEEE

[1]A. Sürekçi, H. H. Kösal, and M. A. Güngör, “A Note on Gershgorin Disks in the Elliptic Plane”, Journal of Mathematical Sciences and Modelling, vol. 4, no. 3, pp. 104–109, Dec. 2021, doi: 10.33187/jmsm.986344.

ISNAD

Sürekçi, Arzu - Kösal, Hidayet Hüda - Güngör, Mehmet Ali. “A Note on Gershgorin Disks in the Elliptic Plane”. Journal of Mathematical Sciences and Modelling 4/3 (December 1, 2021): 104-109. https://doi.org/10.33187/jmsm.986344.

JAMA

1.Sürekçi A, Kösal HH, Güngör MA. A Note on Gershgorin Disks in the Elliptic Plane. Journal of Mathematical Sciences and Modelling. 2021;4:104–109.

MLA

Sürekçi, Arzu, et al. “A Note on Gershgorin Disks in the Elliptic Plane”. Journal of Mathematical Sciences and Modelling, vol. 4, no. 3, Dec. 2021, pp. 104-9, doi:10.33187/jmsm.986344.

Vancouver

1.Arzu Sürekçi, Hidayet Hüda Kösal, Mehmet Ali Güngör. A Note on Gershgorin Disks in the Elliptic Plane. Journal of Mathematical Sciences and Modelling. 2021 Dec. 1;4(3):104-9. doi:10.33187/jmsm.986344