A COMPUTATION PERSPECTIVE FOR THE EIGENVALUES OF CIRCULANT MATRICES INVOLVING GEOMETRIC PROGRESSION

Authors

  • SISWANDI SISWANDI
  • SUGI GURITMAN
  • NUR ALIATININGTYAS
  • TEDUH WULANDARI

DOI:

https://doi.org/10.25077/jmua.12.1.65-77.2023

Keywords:

Circulant matrix, Eigenvalues, Inverse, Cyclic group, Geometric sequence

Abstract

In this article, the eigenvalues and inverse of circulant matrices with entries in the first row having the form of a geometric sequence are formulated explicitly in a simple form in one theorem. The method for deriving the formulation of the determinant and inverse is simply using elementary row or column operations. For the eigenvalues, the known formulation of the previous results is simplified by considering the specialty of the sequence and using cyclic group properties of unit circles in the complex plane. Then, the algorithm of eigenvalues formulation is constructed, and it shows as a better computation method.

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References

Aliatiningtyas N., Guritman T., Wulandari T., 2022, On the Explicit Formula for Eigenvalues, Determinant, and Inverse of Circulant Matrices, Jurnal Teori dan Aplikasi Matematika, Vol. 6(3), DOI:

https://doi.org/10.31764/jtam.v6i3.8616

Ma J., Qiu T., and He C., 2021, A New Method of Matrix Decomposition

to Get the Determinants and Inverses of r-Circulant Matrices with Fibonacci

and Lucas Numbers, Journal of Mathematics, Hindawi, Vol. 2021: Article ID

, https://doi.org/10.1155/2021/4782594

Wei Y., Zheng Y., Jiang Z., Shon S., 2020, Determinants, inverses, norms,

and spreads of skew circulant matrices involving the product of Fibonacci and

Lucas numbers, Journal Mathematics and Computer Sciences, Vol. 20: 64 –

Bueno, A.C.F., 2020, On r-circulant matrices with Horadam numbers having

arithmetic indices, Notes on Number Theory and Discrete Mathematics, Vol.

(2): 177 – 197

Liu Z., Chen S., Xu W., Zhang Y., 2019, The eigen-structures of real (skew)circulant matrices with some applications, Journal Computational and Applied

Mathematics, Springer, Vol. 38(178)

Radicic B., 2019, On k-Circulant Matrices Involving the Jacobsthal Numbers,

Revista de la Union Matematica Argentina, Vol. 60(2): 431 – 442

Bahs M., Solak S., 2018, On The g-Circulant Matrices, Commun. Korean Math.

Soc., Vol. 33(3): 695 – 704

Bozkurt D., Tam T.-Y., 2016, Determinants and inverses of r-circulant matrices associated with a number sequence, Linear and Multilinear Algebra, Taylor

& Francis Online, Vol. 63(10): 2079 – 2088

Radicic B., 2016, On k-circulant matrices (with geometric sequence), Quaestiones Mathematicae, Taylor & Francis Online, Vol. 39(1): 135 – 144

Jiang X., Hong K., 2015, Explicit inverse matrices of Tribonacci skew circulant

type matrices. Applied Mathematics and Computation, EIsevier, Vol. 268: 93

– 102

Jia J. and Li S., 2015, On the inverse and determinant of general bordered

tridiagonal matrices, Applied Mathematics and Computation, EIsevier, Vol.

(6): 503 – 509

Li J., Jiang Z., and Lu F., 2014, Determinants, Norms, and the Spread of

Circulant Matrices with Tribonacci and Generalized Lucas Numbers, Abstract

and Applied Analysis, Hindawi Publishing Corporation, Vol. 2014: Article ID

, https://dx.doi.org/10.1155/2014/381829.

Jiang Z. and Li D., 2014, The Invertibility, Explicit Determinants, and Inverses of Circulant and Left Circulant and G-Circulant Matrices Involving

Any Continuous Fibonacci and Lucas Numbers. In Abstract and Applied

Analysis, Hindawi Publishing Corporation, Vol. 2014: Article ID 931451,

https://dx.doi.org/10.1155/2014/931451.

Jiang Z., Gong Y., and Gao Y., 2014, Invertibility and Explicit Inverses of

Circulant-Type Matrices with k-Fibonacci and k-Lucas Numbers. In Abstract

and Applied Analysis, Hindawi Publishing Corporation, Vol. 2014: Article ID

, https://dx.doi.org/10.1155/2014/238953.

Bueno A. C. F., 2012, Right Circulant Matrices With Geometric Progression,

International Journal of Applied Mathematical Research, Vol. 1(4): 593 – 603

Shen S.-Q., Cen J.-M., and Hao Y., 2011On the determinants and inverses of

circulant matrices with Fibonacci and Lucas numbers. Applied Mathematics

and Computation, EIsevier, Vol. 217(23): 9790 – 9797

Aldrovandi R., 2011, Special Matrices of Mathematical Physics: Stochastic,

Circulant and Bell Matrices, World Scientific, Singapore.

Grimaldi R. P., 1999, Discrete and Combinatorial Mathematics, 4th Edition,

North-Holland Mathematical Library, Addison Wesley Longman Inc.

Lancaster P. and Tismenetski M., 1985, The Theory of Matrices, 2nd ed.,

Academic Press Inc.

Davis P. J., 1979, Circulant Matrices, Wiley, New York

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Published

30-01-2023

How to Cite

SISWANDI, S., GURITMAN, S., ALIATININGTYAS, N., & WULANDARI, T. (2023). A COMPUTATION PERSPECTIVE FOR THE EIGENVALUES OF CIRCULANT MATRICES INVOLVING GEOMETRIC PROGRESSION. Jurnal Matematika UNAND, 12(1), 65–77. https://doi.org/10.25077/jmua.12.1.65-77.2023

Issue

Vol. 12 No. 1 (2023)

Section

Articles

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