On the Explicit Formula for Eigenvalues, Determinant, and Inverse of Circulant Matrices

Nur Aliatiningtyas; Sugi Guritman; Teduh Wulandari

doi:10.31764/jtam.v6i3.8616

On the Explicit Formula for Eigenvalues, Determinant, and Inverse of Circulant Matrices

Nur Aliatiningtyas, Sugi Guritman, Teduh Wulandari

Abstract

Determining eigenvalues, determinants, and inverse for a general matrix is computationally hard work, especially when the size of the matrix is large enough. But, if the matrix has a special type of entry, then there is an opportunity to make it much easier by giving its explicit formulation. In this article, we derive explicit formulas for determining eigenvalues, determinants, and inverses of circulant matrices with entries in the first row of those matrices in any formation of a sequence of numbers. The main method of our study is exploiting the circulant property of the matrix and associating it with cyclic group theory to get the results of the formulation. In every discussion of those concepts, we also present some computation remarks.

Keywords

Circulant matrix; Eigenvalue; Determinant; Inverse; Cyclic group.

Full Text:

DOWNLOAD [PDF]

References

A. C. F. Bueno. (2020). On r-circulant matrices with Horadam numbers having arithmetic indices. Notes on Number Theory and Discrete Mathematics, 26, 177–197.

Aldous Cesar F. Bueno. (2012). Right circulant matrices with geometric progression. International Journal of Applied Mathematical Research, 1(4), 593–603.

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

B. Radicic. (2019). On k-Circulant Matrices Involving the Jacobsthal Numbers. In Revista de La Union Matematica Argentina, 60(2), 431–442.

Biljana Radicic. (2017). On k-Circulant Matrices with Arithmetic Sequence. Filomat, University of Nis, Serbiavol, 31(8), 2517–2525.

Biljana Radicic. (2018a). On k-Circulant Matrices with the Lucas Numbers. Filomat, Faculty of Sciences and Mathematics, University of Nis, Serbiavol, 32(11), 4037–4046.

Biljana Radicic. (2018b). On the Determinants and Inverses of r-Circulant Matrices with the Biperiodic Fibonacci and Lucas Numbers. Filomat, University of Nis, Serbiavol, 32(11), 3637–3650.

Biljana Radicic. (2019). On k-Circulant Matrices Involving the Jacobsthal Numbers. Revista de La Union Matematica Argentina, 60(2), 431–442.

D. Bozkurt, & T.-Y. Tam. (2016). Determinants and inverses of r-circulant matrices associated with a number sequence. Linear and Multilinear Algebra, Taylor & Francis Online, 63(10), 2079–2088.

Emrullah Kirklar, & Fatih Yilmaz. (2019). A General Formula for Determinants and Inverses of r-Ciruculant Matrices with Third Order Recurrences. Mathematical Sciences and Applications E-Notes, 7(2), 1–8.

Ercan Altinisik, N.Feyza yalcin, & Serife Buyukkose. (2015). Determinans and invers of circulant matrices with complex Fibonacci numbers. Spec. Matrices, DE GRUYTER, 3, 82–90.

Fuyong, L. (2011). The inverse of circulant matrix. Applied Mathematics and Computation, 217(21), 8495–8503. https://doi.org/10.1016/j.amc.2011.03.052

Jiang, Z., Gong, Y., & Gao, Y. (2014). Invertibility and Explicit Inverses of Circulant-Type Matrices with k-Fibonacci and k-Lucas Numbers. Abstract and Applied Analysis, 2014, 1–9. https://doi.org/10.1155/2014/238953

Jiang, Z., Wang, W., Zheng, Y., Zuo, B., & Niu, B. (2019). Interesting Explicit Expressions of Determinants and Inverse Matrices for Foeplitz and Loeplitz Matrices. Mathematics, 7(10), 939. https://doi.org/10.3390/math7100939

Jiang, Z., Yao, J., & Lu, F. (2014). On Skew Circulant Type Matrices Involving Any Continuous Fibonacci Numbers. Abstract and Applied Analysis, 2014, 1–10. https://doi.org/10.1155/2014/483021

Jinjiang Yao, & Jixiu Sun. (2018). Explicit Determinants and Inverses of Skew Circulant and Skew Left Circulant Matrices with the Pell-Lucas Numbers. Journal of Advances in Mathematics and Computer Science, 26(1), 1–16.

Jiteng Jia, & Sumei Li. (2015). On the inverse and determinant of general bordered tridiagonal matrices. 69(6), 503–509.

Ma, J., Qiu, T., & 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, 2021, 1–9. https://doi.org/10.1155/2021/4782594

Mustafa Bahsi, & Soleyman Solak. (2018). On The g-Circulant Matrices . Commun. Korean Math. Soc., 33(3), 695–704.

Nazmiye Yilmaz, Yasin Yazlik, & Necaati Taskara. (2016). On the g-Circulant Matrix involving the Generalized k-Horadam Numbers.

Türkmen, R., & Gökbaş, H. (2016). On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers. Journal of Inequalities and Applications, 2016(1), 65. https://doi.org/10.1186/s13660-016-0997-0

X. Jiang, & K. Hong. (2015). Explicit inverse matrices of Tribonacci skew circulant type matrices. Applied Mathematics and Computation, EIsevier, 268(October), 93–102.

Xiaoting Chen. (2019). Determinants and Inverses of Skew Symmetric Generalized Foeplits Matrices. Journal of Advances in Mathematics and Computer Science, 33(4), 1–12.

Yun Fun, & Hualu Liu. (2018). Double Circulant Matrices, Linear and Multilinear Algebra. Taylor & Francis Online, 66(19), 2119–2137.

Yunlan Wei, Yanpeng Zheng, Zhaolin Jiang, & Sugoog Shon. (2020). Determinants, inverse, norms and spreads of skew circulant matrices involving the product of Fibonacci and Lucas Numbers. Journal Mathematics and Computer Sciences, 20, 64–78.

Zhongyun Liu, Siheng Chen, Weijin Xu, & Yulin Zhang. (2019). The eigen-structures of real (skew) circulant matrices with some application. Journal Computational and Applied Mathmatics, Springer, 38(178).

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