Determinant evaluations for binary circulant matrices
Christos Kravvaritis
Special Matrices, Tome 2 (2014), / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 2014-01-01
EUDML-ID : urn:eudml:doc:269493
@article{bwmeta1.element.doi-10_2478_spma-2014-0019,
     author = {Christos Kravvaritis},
     title = {Determinant evaluations for binary circulant matrices},
     journal = {Special Matrices},
     volume = {2},
     year = {2014},
     zbl = {1310.15012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0019}
}
Christos Kravvaritis. Determinant evaluations for binary circulant matrices. Special Matrices, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0019/

[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] | Zbl 0533.15007

[3] B. Fischer and J. Modersitzki. Fast inversion of matrices arising in image processing. Numer. Algorithms 22:1–11, 1999. [Crossref] | Zbl 0957.65019

[4] S. Georgiou and C. Kravvaritis. New Good Quasi-Cyclic Codes over GF(3). Int. J. Algebra 1:11–24, 2007. | Zbl 1206.94106

[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. | Zbl 0496.15002

[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. | Zbl 1110.65034

[8] K. J. Horadam. Hadamard matrices and their applications. Princeton University Press, Princeton and Oxford, 2007. | Zbl 1145.05014

[9] R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, Cambridge, 1985. | Zbl 0576.15001

[10] T. K. Huckle. Compact Fourier Analysis for Designing Multigrid Methods. SIAM J. Comput. 31:644–666, 2008. [WoS] | Zbl 1186.65037

[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] | Zbl 1250.65151

[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. | Zbl 0947.65031

[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. | Zbl 0981.65056

[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. | Zbl 0788.62067

[15] C. Krattenthaler. Advanced determinant calculus. Sém. Lothar. Combin. 42:69–157, 1999. | Zbl 0923.05007

[16] C. Krattenthaler. Advanced determinant calculus: A complement. Linear Algebra Appl., 411:68–166, 2005. | Zbl 1079.05008

[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. | Zbl 1146.15004

[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] | Zbl 1040.68567

[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] | Zbl 1048.15008

[21] F. R. Sharpe. The maximum value of a determinant. Bull. Amer. Math. Soc. 14:121–123, 1907. [Crossref] | Zbl 38.0200.03