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Year 2016, Volume 29, Issue 2, 487 - 490, 21.06.2016

Abstract

References

  • Altınışık, E., Yalçın, N.F. and Büyükköse, Ş., “Determinants and inverses of circulant matrices with complex Fibonacci numbers”, Special Matrices, 3:82-90 (2015).
  • Atkin A.O.L., Boros,E., Cechlarova, K.U. and N. Peled, “Powers of Circulants in Bottlenack Algebra”, Linear Algebra and Its Appl., 258: 137-148 (1997).
  • Bahsi, M. and Solak, S., “On the circulant matrices with arithmetic sequence”, Int. J. Cont. Math. Sciences, 5(25): 1213 – 1222 (2010).
  • Bose, A. and Mitra, J., “Limiting Spectral Distribution of a Special Circulant”, Statistics & Prob. Let., 60: 111-120 (2002).
  • Civciv, H. and Türkmen, R., “Notes on norms of circulant matrices with Lucas number”, Int. Journal for Inf. and System Sci., 4(1): 142-147 (2008).
  • Davis, P.J., Circulant Matrices, Wiley, New York, Chichester, Brisbane, 1979.
  • Hladnik, M., “Schur Norms of Bicirculant Matrices”, Linear Algebra and Its Appl., 286: 261-272 (1999).
  • Horadam, A.F., “Complex Fibonacci numbers and Fibonacci quaternions”, Amer. Math. Monthly 70: 289-291 (1963).
  • Horn, R.A., Johnson, C.R., Matrix Analysis, Cambridge University Press, Cambridge, 1985.
  • Jiang, Z., Xin, H. and Lu, F., “Gaussian Fibonacci circulant type matrices”, Abstr. Appl. Anal., Art. ID 592782, 10 pp, (2014).
  • Ipek, A., “On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries”, Appl. Math. Comp. 217: 6011-6012 (2011).
  • Karner, H., Schneid, J., and Ueberhuber, C.W., “Spectral Decomposition of Real Circulant Matrices”, Linear Algebra and Its Appl., 367: 301-311 (2003).
  • Shen, S. and Cen, J., “On the bounds for the norms of r- Circulant Matrices with the Fibonacci and Lucas Numbers”, Appl. Math. Comput., 216: 2891-2897 (2010).
  • Shen, S.Q., Cen, J.M. and Hao, Y., “On the determinants and inverses of circulant matrices with and Lucas numbers”, Appl. Math. Comput., 217: 9790-9797 (2011).
  • Solak, S., “On the norms of circulant matrices with the Fibonacci and Lucas numbers”, Appl. Math. Comp. 160: 125-132 (2005).
  • Solak, S., Erratum to "On the Norms of Circulant Matrices with the Fibonacci and Lucas Numbers" [Appl. Math. Comp., 160, (2005), 125-132], Appl. Math. Comp., 190: 1855-1856 (2007).
  • Solak, S. and Bozkurt, D., “Some bounds on matrix and operator norms of almost circulant, Cauchy-Toeplitz and Cauchy-Hankel matrices”, Math. Comp. Appl. Int. J., 7(3): 211-218 (2002).
  • Tuğlu, N. and Kızılateş C., “On the Norms of Some Special Matrices with the Harmonic Fibonacci Numbers”, Gazi University Journal of Science, 28(3): 497-501 (2015).
  • Vajda, S., Fibonacci & Lucas Numbers, and the Golden Section. West Sussex, England: Ellis Horwood Limited, 1989.
  • Zhang, S., Jiang, Z. and Liu, S., “An Application of the Gröbner Basis in Computation for the Minimal Polynomials and Inverses of Block Circulant Matrices”, Linear Algebra and Its Appl., 347: 101-114 (2002).

ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS

Year 2016, Volume 29, Issue 2, 487 - 490, 21.06.2016

Abstract

In this paper, we compute the norms of circulant matrices with the complex Fibonacci and Lucas numbers. Moreover, we give golden ratio in complex Fibonacci numbers.In some scientific areas such as signal processing, coding theory and image processing, we often encounter circulant matrices. An n n  matrix C is called a circulant matrix if it is of the form 0 1 1 1 0 2 2 1 3 1 2 0 n n n n n c c c c c c C c c c c c c            

  
 
 
   
or an n n 
matrix C is circulant if there exist
0 1 1 , , ,
n
c c c 
such that the i, j entry of C is
 j i n mod c 
, where the rows and columns are numbered
from 0 to
n 1
and kmodn means the number between 0
to
n 1
that is congruent to kmodn. Thus, we denote the
circulant matrix C as C Circ c c c   0 1 1 , , ,
n 
. Any
circulant matrix has many elegant properties. Some of
them are [6,12]

References

  • Altınışık, E., Yalçın, N.F. and Büyükköse, Ş., “Determinants and inverses of circulant matrices with complex Fibonacci numbers”, Special Matrices, 3:82-90 (2015).
  • Atkin A.O.L., Boros,E., Cechlarova, K.U. and N. Peled, “Powers of Circulants in Bottlenack Algebra”, Linear Algebra and Its Appl., 258: 137-148 (1997).
  • Bahsi, M. and Solak, S., “On the circulant matrices with arithmetic sequence”, Int. J. Cont. Math. Sciences, 5(25): 1213 – 1222 (2010).
  • Bose, A. and Mitra, J., “Limiting Spectral Distribution of a Special Circulant”, Statistics & Prob. Let., 60: 111-120 (2002).
  • Civciv, H. and Türkmen, R., “Notes on norms of circulant matrices with Lucas number”, Int. Journal for Inf. and System Sci., 4(1): 142-147 (2008).
  • Davis, P.J., Circulant Matrices, Wiley, New York, Chichester, Brisbane, 1979.
  • Hladnik, M., “Schur Norms of Bicirculant Matrices”, Linear Algebra and Its Appl., 286: 261-272 (1999).
  • Horadam, A.F., “Complex Fibonacci numbers and Fibonacci quaternions”, Amer. Math. Monthly 70: 289-291 (1963).
  • Horn, R.A., Johnson, C.R., Matrix Analysis, Cambridge University Press, Cambridge, 1985.
  • Jiang, Z., Xin, H. and Lu, F., “Gaussian Fibonacci circulant type matrices”, Abstr. Appl. Anal., Art. ID 592782, 10 pp, (2014).
  • Ipek, A., “On the spectral norms of circulant matrices with classical Fibonacci and Lucas numbers entries”, Appl. Math. Comp. 217: 6011-6012 (2011).
  • Karner, H., Schneid, J., and Ueberhuber, C.W., “Spectral Decomposition of Real Circulant Matrices”, Linear Algebra and Its Appl., 367: 301-311 (2003).
  • Shen, S. and Cen, J., “On the bounds for the norms of r- Circulant Matrices with the Fibonacci and Lucas Numbers”, Appl. Math. Comput., 216: 2891-2897 (2010).
  • Shen, S.Q., Cen, J.M. and Hao, Y., “On the determinants and inverses of circulant matrices with and Lucas numbers”, Appl. Math. Comput., 217: 9790-9797 (2011).
  • Solak, S., “On the norms of circulant matrices with the Fibonacci and Lucas numbers”, Appl. Math. Comp. 160: 125-132 (2005).
  • Solak, S., Erratum to "On the Norms of Circulant Matrices with the Fibonacci and Lucas Numbers" [Appl. Math. Comp., 160, (2005), 125-132], Appl. Math. Comp., 190: 1855-1856 (2007).
  • Solak, S. and Bozkurt, D., “Some bounds on matrix and operator norms of almost circulant, Cauchy-Toeplitz and Cauchy-Hankel matrices”, Math. Comp. Appl. Int. J., 7(3): 211-218 (2002).
  • Tuğlu, N. and Kızılateş C., “On the Norms of Some Special Matrices with the Harmonic Fibonacci Numbers”, Gazi University Journal of Science, 28(3): 497-501 (2015).
  • Vajda, S., Fibonacci & Lucas Numbers, and the Golden Section. West Sussex, England: Ellis Horwood Limited, 1989.
  • Zhang, S., Jiang, Z. and Liu, S., “An Application of the Gröbner Basis in Computation for the Minimal Polynomials and Inverses of Block Circulant Matrices”, Linear Algebra and Its Appl., 347: 101-114 (2002).

Details

Primary Language English
Subjects Engineering
Journal Section Mathematics
Authors

Süleyman Solak


Mustafa Bahşi
0000-0002-6356-6592

Publication Date June 21, 2016
Published in Issue Year 2016, Volume 29, Issue 2

Cite

Bibtex @conference paper { gujs259007, journal = {Gazi University Journal of Science}, issn = {}, eissn = {2147-1762}, address = {}, publisher = {Gazi University}, year = {2016}, volume = {29}, pages = {487 - 490}, doi = {}, title = {ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS}, key = {cite}, author = {Solak, Süleyman and Bahşi, Mustafa} }
APA Solak, S. & Bahşi, M. (2016). ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS . Gazi University Journal of Science , 29 (2) , 487-490 . Retrieved from https://dergipark.org.tr/en/pub/gujs/issue/24433/259007
MLA Solak, S. , Bahşi, M. "ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS" . Gazi University Journal of Science 29 (2016 ): 487-490 <https://dergipark.org.tr/en/pub/gujs/issue/24433/259007>
Chicago Solak, S. , Bahşi, M. "ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS". Gazi University Journal of Science 29 (2016 ): 487-490
RIS TY - JOUR T1 - ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS AU - Süleyman Solak , Mustafa Bahşi Y1 - 2016 PY - 2016 N1 - DO - T2 - Gazi University Journal of Science JF - Journal JO - JOR SP - 487 EP - 490 VL - 29 IS - 2 SN - -2147-1762 M3 - UR - Y2 - 2021 ER -
EndNote %0 Gazi University Journal of Science ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS %A Süleyman Solak , Mustafa Bahşi %T ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS %D 2016 %J Gazi University Journal of Science %P -2147-1762 %V 29 %N 2 %R %U
ISNAD Solak, Süleyman , Bahşi, Mustafa . "ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS". Gazi University Journal of Science 29 / 2 (June 2016): 487-490 .
AMA Solak S. , Bahşi M. ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS. Gazi University Journal of Science. 2016; 29(2): 487-490.
Vancouver Solak S. , Bahşi M. ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS. Gazi University Journal of Science. 2016; 29(2): 487-490.
IEEE S. Solak and M. Bahşi , "ON THE NORMS OF CIRCULANT MATRICES WITH THE COMPLEX FIBONACCI AND LUCAS NUMBERS", Gazi University Journal of Science, vol. 29, no. 2, pp. 487-490, Jun. 2016