DURMUŞ BOZKURT, CARLOS M. DAFONSECA, FATİHYILMAZ

THE DETERMINANTS OF CIRCULANT AND SKEW-CIRCULANT MATRICES WITH TRIBONACCI NUMBERS

Determinant computation has an important role in mathematics.It can be computed by using some different methods but it needs huge amountof operations to compute determinant of a matrix. For instance, using Gausselimination method, it is neccessary about 2n3/3 arithmetic steps of a matrix of order n. Therefore, determinant computation has been considered forspecial matrices with special entries. Similarly, the permanent of a matrix isan analog of determinant where all the signs in the expansion by minors aretaken as positive. This study considers the determinant of circulant matriceswhose entries are Tribonacci numbers. Some relations with the permanent areestablished

Keywords:

Circulant matrix; Tribonacci number; Determinant; Permanent,

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D. Bozkurt, T.-Y. Tam, Determinants and Inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers, Appl. Math. Comput. 219 (2012), no.2, 544-551.
A. Dasdemir, On the Pell, Pell-Lucas and modified Pell numbers by matrix method. Appl. Math. Sci. (Ruse) 5 (2011), no. 61-64, 3173-3181.
P.J. Davis, Circulant Matrices, Wiley, NewYork, 1979.
C.M. da Fonseca, An identity between the determinant and the permanent of Hessenberg-type matrices, Czechoslovak Math. J. 61(136) (2011), no.4, 917-921.
C.M. da Fonseca, On the location of the eigenvalues of Jacobi matrices, Appl. Math. Lett. 19 (2006), no.11, 1168-1174.
P.M. Gibson, An identity between permanents and determinants, Amer. Math. Monthly 76 (1969), 270-271.
H. Karner, J. Schneid, C.W. Ueberhuber, Spectral decomposition of real circulant matrices, Linear Algebra Appl. 367 (2003), 301-311.
T. Koshy, Fibonacci and Lucas Numbers with Applications, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2001.
G. Y. Lee, k-Lucas numbers and associated bipartite graphs, Linear Algebra Appl. 320 (2000), no.1-3, 51-61.
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), no.23, 9790-9797.
N.J.A. Sloane, The on-line encyclopedia of integer sequences, 1999, http://oeis.org/A000073
F. Yılmaz, D. Bozkurt, Hessenberg matrices and the Pell and Perrin numbers, J. Number Theory 131 (2011), no.8, 1390-1396.
F. Zhang, Matrix Theory, Basic Results and Techniques, Springer, New York, 2011.
G. Zhao, The improved nonsingularity on the r-circulant matrices in signal processing, In- ternational Conference on Computer Technology and Development - ICCTD 2009, Kota Kinabalu, 564-567.
E. Gokcen Alptekin, T. Mansour,and N. Tuglu, Norms of Circulant and Semicirculant ma- trices and Horadam’s sequence, Ars combinatorica 85 (2007) 353–359.
R.A. Brualdi, P.M. Gibson, Convex polyhedra of doubly stochastic matrices I: applications of the permanent function, J. Combin. Theory A 22 (1977), 194-230.
Department of Mathematics, Selcuk University, 42250 Konya, Turkey.
E-mail address: dbozkurt@selcuk.edu.tr Department of Mathematics, Kuwait University, Safat 13060, Kuwait.
E-mail address: carlos@sci.kuniv.edu.kw Department of Mathematics, Gazi University, 06900 Ankara, Turkey.
E-mail address: fatihyilmaz@gazi.edu.tr