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Tytuł artykułu

Determinant evaluations for binary circulant matrices

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Języki publikacji

EN

Abstrakty

EN
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.

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Wydawca

Czasopismo

Rocznik

Tom

2

Numer

1

Opis fizyczny

Daty

otrzymano
2014-05-28
zaakceptowano
2014-11-13
online
2014-12-09

Twórcy

  • Department of Mathematics, University of Athens, Panepistimiopolis 15784,
    Athens, Greece

Bibliografia

  • [1] J. Bae. Circulant Matrix Factorization Based on Schur Algorithm for Designing Optical Multimirror Filters. Japan. J. Math. 45:5163–5168, 2006.
  • [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. [WoS]
  • [3] B. Fischer and J. Modersitzki. Fast inversion of matrices arising in image processing. Numer. Algorithms 22:1–11, 1999. [Crossref]
  • [4] S. Georgiou and C. Kravvaritis. New Good Quasi-Cyclic Codes over GF(3). Int. J. Algebra 1:11–24, 2007.
  • [5] R. M. Gray. Toeplitz and Circulant Matrices: A review. Found. Trends Comm. Inform. Theory 2:155–239, 2006.
  • [6] F. A. Graybill. Matrices with applications in statistics. Prentice Hall, Wadsworth-Belmont, 1983.
  • [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.
  • [8] K. J. Horadam. Hadamard matrices and their applications. Princeton University Press, Princeton and Oxford, 2007.
  • [9] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985.
  • [10] T. K. Huckle. Compact Fourier Analysis for Designing Multigrid Methods. SIAM J. Comput. 31:644–666, 2008. [WoS]
  • [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. [WoS]
  • [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.
  • [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.
  • [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.
  • [15] C. Krattenthaler. Advanced determinant calculus. Sém. Lothar. Combin. 42:69–157, 1999.
  • [16] C. Krattenthaler. Advanced determinant calculus: A complement. Linear Algebra Appl., 411:68–166, 2005.
  • [17] G. Maze and H. Parlier. Determinants of Binary Circulant matrices. IEEE Trans. Inform. Theory p. 124, 2004.
  • [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.
  • [19] N. Nguyen, P. Milanfar and G. Golub. A Computationally Eflcient Superresolution Image Reconstruction Algorithm. IEEE Trans. Image Process. 10:573–583, 2001. [Crossref][PubMed]
  • [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. [WoS]
  • [21] F. R. Sharpe. The maximum value of a determinant. Bull. Amer. Math. Soc. 14:121–123, 1907. [Crossref]

Typ dokumentu

Bibliografia

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bwmeta1.element.doi-10_2478_spma-2014-0019
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