Research article

Determinants and invertibility of circulant matrices

  • Received: 18 June 2024 Revised: 16 July 2024 Accepted: 19 July 2024 Published: 30 July 2024
  • Let $ a_0, a_1, \dots, a_{n-1} $ be real numbers and let $ A = Circ(a_0, a_1, \dots, a_{n-1}) $ be a circulant matrix with $ f(x) = \Sigma ^{n-1}_{j = 0}a_jx^j $. First, we prove that $ Circ(a_0, a_1, \dots, a_{n-1}) $ must be invertible if the sequence $ a_0, a_1, \dots, a_{n-1} $ is a strictly monotonic sequence and $ a_0+a_1+\dots+a_{n-1}\neq 0 $. Next, we reduce the calculation of $ f(\varepsilon ^0)f(\varepsilon)\dots f(\varepsilon ^{n-1}) $ for a prime $ n $ by using the techniques on finite fields, where $ \varepsilon $ is a primitive $ n $-th root of unity. Finally, we provide two examples to explain how to use the obtained results to calculate the determinant of a circulant matrix.

    Citation: Xiuyun Guo, Xue Zhang. Determinants and invertibility of circulant matrices[J]. Electronic Research Archive, 2024, 32(7): 4741-4752. doi: 10.3934/era.2024216

    Related Papers:

  • Let $ a_0, a_1, \dots, a_{n-1} $ be real numbers and let $ A = Circ(a_0, a_1, \dots, a_{n-1}) $ be a circulant matrix with $ f(x) = \Sigma ^{n-1}_{j = 0}a_jx^j $. First, we prove that $ Circ(a_0, a_1, \dots, a_{n-1}) $ must be invertible if the sequence $ a_0, a_1, \dots, a_{n-1} $ is a strictly monotonic sequence and $ a_0+a_1+\dots+a_{n-1}\neq 0 $. Next, we reduce the calculation of $ f(\varepsilon ^0)f(\varepsilon)\dots f(\varepsilon ^{n-1}) $ for a prime $ n $ by using the techniques on finite fields, where $ \varepsilon $ is a primitive $ n $-th root of unity. Finally, we provide two examples to explain how to use the obtained results to calculate the determinant of a circulant matrix.



    加载中


    [1] I. B. Collings, I. Vaughan, L. Clarkson, A low-complexity lattice-based low-PAR transmission scheme for DSL channels, IEEE Trans. Commun., 52 (2004), 755–764. https://doi.org/10.1109/TCOMM.2004.826261 doi: 10.1109/TCOMM.2004.826261
    [2] P. J. Davis, Circulant Matrices, $2^{nd}$ edition, Chelsea Publishing, New York, 1994.
    [3] B. Gellai, Determination of molecular symmetry coordinates using circulant matrices, J Mol. Struct., 1 (1984), 21–26. https://doi.org/10.1016/S0022-2860(84)87196-3 doi: 10.1016/S0022-2860(84)87196-3
    [4] A. Carmona, A. M. Encinas, S. Gagoa, M. J. Jiménez, M. Mitjana, The inverses of some circulant matrices, Appl. Math. Comput., 270 (2015), 785–793. https://doi.org/10.1016/j.amc.2015.08.084 doi: 10.1016/j.amc.2015.08.084
    [5] F. Lin, The inverse of circulant matrix, Appl. Math. Comput., 217 (2011), 8495–8503. https://doi.org/10.1016/j.amc.2011.03.052 doi: 10.1016/j.amc.2011.03.052
    [6] O. Rojo, H. Rojo, Some results on symmetric circulant matrices and on symmetric centrosymmetric matrices, Linear Algebra Appl., 392 (2004), 211–233. https://doi.org/10.1016/j.laa.2004.06.013 doi: 10.1016/j.laa.2004.06.013
    [7] S. Q. Shen, J. M. Cen, Y. Hao, On the determinants and inverses of circulant matrices with fibonacci and lucas numbers, Appl. Math. Comput., 217 (2011), 9790–9797. https://doi.org/10.1016/j.amc.2011.04.072 doi: 10.1016/j.amc.2011.04.072
    [8] R. M. Gray, Toeplitz and circulant matrices: A review,, Found. Trends Commun. Inf. Theory, 2 (2006), 155–239. http://doi.org/10.1561/0100000006 doi: 10.1561/0100000006
    [9] M. N. Huxley, M. C. Lettington, K. M. Schmidt, On the structure of additive systems of integers, Period. Math. Hung., 78 (2019), 178–199. http://doi.org/10.1007/s10998-018-00275-w doi: 10.1007/s10998-018-00275-w
    [10] M. C. Lettington, K. M. Schmidt, Divisor functions and the number of sum systems, Integers, preprint, arXiv: 1910.02455. https://doi.org/10.48550/arXiv.1910.02455
    [11] M. C. Lettington, K. M. Schmidt, On the sum of left and right circulant matrices, Linear Algebra Appl., 658 (2023), 62–85. https://doi.org/10.1016/j.laa.2022.10.024 doi: 10.1016/j.laa.2022.10.024
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(460) PDF downloads(35) Cited by(0)

Article outline

Figures and Tables

Figures(2)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog