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Special Matrices

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Determinant evaluations for binary circulant matrices

Christos Kravvaritis
Published Online: 2014-12-09 | DOI: https://doi.org/10.2478/spma-2014-0019

Abstract

Determinant formulas for special binary circulant matrices are derived and a new open problem regarding the possible determinant values of these specific circulant matrices is stated. The ideas used for the proofs can be utilized to obtain more determinant formulas for other binary circulant matrices, too. The superiority of the proposed approach over the standard method for calculating the determinant of a general circulant matrix is demonstrated.

Keywords: Determinant; binary circulant matrices

MSC: 15A15; 65F40

References

  • [1] J. Bae. Circulant Matrix Factorization Based on Schur Algorithm for Designing Optical Multimirror Filters. Japan. J. Math. 45:5163–5168, 2006. Google Scholar

  • [2] R. A. Brualdi and H. Schneider. Determinantal identities: Gauss, Schur, Cauchy, Sylvester, Kronecker, Jacobi, Binet, Laplace, Muir and Cayley. Linear Algebra Appl. 52:769–791, 1983. Web of ScienceGoogle Scholar

  • [3] B. Fischer and J. Modersitzki. Fast inversion of matrices arising in image processing. Numer. Algorithms 22:1–11, 1999. CrossrefGoogle Scholar

  • [4] S. Georgiou and C. Kravvaritis. New Good Quasi-Cyclic Codes over GF(3). Int. J. Algebra 1:11–24, 2007. Google Scholar

  • [5] R. M. Gray. Toeplitz and Circulant Matrices: A review. Found. Trends Comm. Inform. Theory 2:155–239, 2006. Google Scholar

  • [6] F. A. Graybill. Matrices with applications in statistics. Prentice Hall, Wadsworth-Belmont, 1983. Google Scholar

  • [7] K. Grifln and M. J. Tsatsomeros. Principal minors, Part I: A method for computing all the principal minors of amatrix. Linear Algebra Appl. 419:107–124, 2006. Google Scholar

  • [8] K. J. Horadam. Hadamard matrices and their applications. Princeton University Press, Princeton and Oxford, 2007. Google Scholar

  • [9] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985. Google Scholar

  • [10] T. K. Huckle. Compact Fourier Analysis for Designing Multigrid Methods. SIAM J. Comput. 31:644–666, 2008. Web of ScienceGoogle Scholar

  • [11] T. K. Huckle and C. Kravvaritis, Compact Fourier Analysis for Multigrid Methods based on Block Symbols, SIAM J. Matrix Anal. Appl., 33:73–96, 2012. Web of ScienceGoogle Scholar

  • [12] C. Koukouvinos, M. Mitrouli and J. Seberry.Growth in Gaussian elimination for weighingmatrices,W(n, n−1). Linear Algebra Appl. 306:189–202, 2000. Google Scholar

  • [13] C. Koukouvinos, M. Mitrouli and J. Seberry. An algorithm to find formulae and values of minors for Hadamard matrices. Linear Algebra Appl., 330:129–147, 2001. Google Scholar

  • [14] S. Kounias, C. Koukouvinos, N. Nikolaou and A. Kakos. The nonequivalent circulant D-optimal designs for n ≡ 2mod 4, n = 54, n = 66. J. Combin. Theory Ser. A 65:26–38, 1994. Google Scholar

  • [15] C. Krattenthaler. Advanced determinant calculus. Sém. Lothar. Combin. 42:69–157, 1999. Google Scholar

  • [16] C. Krattenthaler. Advanced determinant calculus: A complement. Linear Algebra Appl., 411:68–166, 2005. Google Scholar

  • [17] G. Maze and H. Parlier. Determinants of Binary Circulant matrices. IEEE Trans. Inform. Theory p. 124, 2004. Google Scholar

  • [18] A. R. Moghaddamfar, S. M. H. Pooya, S. Navid Salehy and S. Nima Salehy. More calculations on determinant evaluations. Electron. J. Linear Algebra 16:19–29, 2007. Google Scholar

  • [19] N. Nguyen, P. Milanfar and G. Golub. A Computationally Eflcient Superresolution Image Reconstruction Algorithm. IEEE Trans. Image Process. 10:573–583, 2001. PubMedCrossrefGoogle Scholar

  • [20] J. Seberry, T. Xia, C. Koukouvinos and M. Mitrouli. The maximal determinant and subdeterminants of ±1 matrices. Linear Algebra Appl. 373:297–310, 2003. Web of ScienceGoogle Scholar

  • [21] F. R. Sharpe. The maximum value of a determinant. Bull. Amer. Math. Soc. 14:121–123, 1907. CrossrefGoogle Scholar

About the article

Received: 2014-05-28

Accepted: 2014-11-13

Published Online: 2014-12-09


Citation Information: Special Matrices, Volume 2, Issue 1, ISSN (Online) 2300-7451, DOI: https://doi.org/10.2478/spma-2014-0019.

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© 2014 Christos Kravvaritis. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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