Determinant evaluations for binary circulant matrices

Christos Kravvaritis

Special Matrices (2014)

  • Volume: 2, Issue: 1, page 187-199, electronic only
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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Christos Kravvaritis. "Determinant evaluations for binary circulant matrices." Special Matrices 2.1 (2014): 187-199, electronic only. <http://eudml.org/doc/269493>.

@article{ChristosKravvaritis2014,
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.},
author = {Christos Kravvaritis},
journal = {Special Matrices},
keywords = {Determinant; binary circulant matrices; determinant},
language = {eng},
number = {1},
pages = {187-199, electronic only},
title = {Determinant evaluations for binary circulant matrices},
url = {http://eudml.org/doc/269493},
volume = {2},
year = {2014},
}

TY - JOUR
AU - Christos Kravvaritis
TI - Determinant evaluations for binary circulant matrices
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 187
EP - 199, electronic only
AB - 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.
LA - eng
KW - Determinant; binary circulant matrices; determinant
UR - http://eudml.org/doc/269493
ER -

References

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  1. [1] J. Bae. Circulant Matrix Factorization Based on Schur Algorithm for Designing Optical Multimirror Filters. Japan. J. Math. 45:5163–5168, 2006. 
  2. [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] Zbl0533.15007
  3. [3] B. Fischer and J. Modersitzki. Fast inversion of matrices arising in image processing. Numer. Algorithms 22:1–11, 1999. [Crossref] Zbl0957.65019
  4. [4] S. Georgiou and C. Kravvaritis. New Good Quasi-Cyclic Codes over GF(3). Int. J. Algebra 1:11–24, 2007. Zbl1206.94106
  5. [5] R. M. Gray. Toeplitz and Circulant Matrices: A review. Found. Trends Comm. Inform. Theory 2:155–239, 2006. 
  6. [6] F. A. Graybill. Matrices with applications in statistics. Prentice Hall, Wadsworth-Belmont, 1983. Zbl0496.15002
  7. [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. Zbl1110.65034
  8. [8] K. J. Horadam. Hadamard matrices and their applications. Princeton University Press, Princeton and Oxford, 2007. Zbl1145.05014
  9. [9] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985. Zbl0576.15001
  10. [10] T. K. Huckle. Compact Fourier Analysis for Designing Multigrid Methods. SIAM J. Comput. 31:644–666, 2008. [WoS] Zbl1186.65037
  11. [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] Zbl1250.65151
  12. [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. Zbl0947.65031
  13. [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. Zbl0981.65056
  14. [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. Zbl0788.62067
  15. [15] C. Krattenthaler. Advanced determinant calculus. Sém. Lothar. Combin. 42:69–157, 1999. Zbl0923.05007
  16. [16] C. Krattenthaler. Advanced determinant calculus: A complement. Linear Algebra Appl., 411:68–166, 2005. Zbl1079.05008
  17. [17] G. Maze and H. Parlier. Determinants of Binary Circulant matrices. IEEE Trans. Inform. Theory p. 124, 2004. 
  18. [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. Zbl1146.15004
  19. [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] Zbl1040.68567
  20. [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] Zbl1048.15008
  21. [21] F. R. Sharpe. The maximum value of a determinant. Bull. Amer. Math. Soc. 14:121–123, 1907. [Crossref] Zbl38.0200.03

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