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On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers

Yıl 2020, Cilt: 12 Sayı: 2, 76 - 85, 31.12.2020
https://doi.org/10.47000/tjmcs.715052

Öz

In this paper, we compute the spectral norms of $r-$ circulant matrices with the hyper-Fibonacci and hyper-Lucas numbers of the forms $F_{r}=Circ-r(F_k^{(0)},F_k^{(1)},...,F_k^{(n-1)}) $$L_r=Circ-r(L_{k}^{\left( 0\right) },L_{k}^{\left( 1\right) },...,L_{k}^{\left( n-1\right) })$ and their Hadamard and Kronecker products. For this, we firstly compute the spectral and Euclidean norms of circulant matrices of the forms $F=Circ(F_{k}^{\left( 0\right) }, F_{k}^{\left( 1\right) },... ,F_{k}^{\left( n-1\right) })$ and $L=Circ(L_{k}^{\left( 0\right) },L_{k}^{\left( 1\right) },...,L_{k}^{\left( n-1\right) })$. Moreover, we give some examples related to special cases of our results.

Kaynakça

  • Bae, J., \textit{Circulant matrix factorization based on schur algorithm for designing optical multimirror filters,} Japanese Journal of Applied Physics \textbf{45(6A)}(2006), 5163--5168.
  • Bah\c{s}i, M., \textit{On the norms of r-circulant matrices with the hyperharmonic numbers}, Journal of Mathematical Inequalities \textbf{10(2)}(2016), 445-458.
  • Bah\c{s}i, M., \textit{On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers,} TWMS J. Pure Appl. Math. \textbf{6(1)}(2015), 84-92.
  • Bah\c{s}i, M., Mez\"{o}, I., Solak, S.,\textit{ A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers}, Annales Mathematicae et Informaticae \textbf{43}(2014), 19--27.
  • Bah\c{s}i, M., S. Solak, \textit{On the norms of r-circulant matrices with the hyper-Fibonacci and Lucas numbers}, Journal of Mathematical Inequalities \textbf{8(4)}(2014), 693-705.
  • Bahsi, M., Solak, S., \textit{On the circulant matrices with arithmetic sequence}, Int. J. Cont. Math. Sciences \textbf{5(25)}(2010), 1213 -- 1222.
  • Cao, N-N., Zhao, F-Z., \textit{Some Properties of Hyperfibonacci and Hyperlucas Numbers,}\ Journal of Integer Sequences \textbf{13}(2010), Article 10.0.8.
  • Davis, P.J., \textit{Circulant Matrices,} Wiley, New York, Chichester, Brisbane, 1979.
  • Dil, A., Mez\"{o}, I., \textit{A symmetric algorithm for hyperharmonic and Fibonacci numbers,}\ Appl. Math. Comp. \textbf{206}(2008), 942-951.
  • Fischer, B., Modersitzki, J., \textit{Fast inversion of matrices arising in image processing,} Numer. Algorithms \textbf{22}(1999), 1--11.
  • Georgiou, S.D., Kravvaritis, C., \textit{New Good Quasi-Cyclic Codes over GF(3),} Int. J. Algebra \textbf{1(1)}(2007), 11--24.
  • Horn, R.A., Johnson, C.R., \textit{Matrix Analysis}, Cambridge University Press, Cambridge, 1985.
  • Horn, R.A., Johnson, C.R., \textit{Topics in Matrix Analysis}, Cambridge University Press, Cambridge, 1991.
  • Karner, H., Schneid, J., Ueberhuber, C.W.,\textit{Spectral Decomposition of Real Circulant Matrices,} Linear Algebra and Its Appl., \textbf{367}(2003), 301-311.
  • K\i z\i late\c{s}, C., Tuglu, N., \textit{On the bounds for the spectral norms of geometric circulant matrices}, Journal of Inequalities and Applications (2016), 2016:312.
  • Kocer, E.G., \textit{Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and Jacobsthal-Lucas numbers,} Hacettepe Journal of Mathematics and Statistics \textbf{36(2)}(2007), 133--142.
  • Kocer, E.G., Mansour, T., Tu\u{g}lu, N., \textit{Norms of circulant and semicirculant matrices with Horadam's numbers,} Ars Combinatoria \textbf{85}(2007), 353-359.
  • Tuglu, N., K\i z\i late\c{s}, C., \textit{On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers}, Journal of Inequalities and Applications (2015), 2015:253.
  • Tuglu, N., K\i z\i late\c{s}, C., \textit{On the Norms of Some Special Matrices with the Harmonic Fibonacci Numbers,} Gazi University Journal of Science \textbf{28(3)}(2015), 497-501.
  • \"{O}cal, A.A., Tu\u{g}lu, N., Alt\i n\i \c{s}\i k, E., \textit{On the representation of k-generalized Fibonacci and Lucas numbers}, Appl. Math. Comp. \textbf{170}(2005), 584-596.
  • Shen, S., Cen, J., \textit{On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers,} Appl. Math. Comp. \textbf{216}(2010), 2891-2897.
  • Solak, S., \textit{On the norms of circulant matrices with the Fibonacci and Lucas numbers}, Appl. Math. Comp. \textbf{160}(2005), 125-132.
  • Solak, S., \textit{Erratum to \textquotedblleft On the Norms of Circulant Matrices with the Fibonacci and Lucas Numbers\textquotedblright \ [Appl. Math. Comp., 160, (2005), 125-132],} Appl. Math. Comp. \textbf{190}(2007), 1855-1856.
  • T\"{u}rkmen, R., G\"{o}kba\c{s}, H., \textit{On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers}, Journal of Inequalities and Applications (2016),2016:65.
  • Yazlik, Y., Taskara, N., \textit{On the norms of an $r-$circulant matrix with the generalized k-Horadam numbers,} Journal of Inequalities and Applications (2013), 2013:394.
Yıl 2020, Cilt: 12 Sayı: 2, 76 - 85, 31.12.2020
https://doi.org/10.47000/tjmcs.715052

Öz

Kaynakça

  • Bae, J., \textit{Circulant matrix factorization based on schur algorithm for designing optical multimirror filters,} Japanese Journal of Applied Physics \textbf{45(6A)}(2006), 5163--5168.
  • Bah\c{s}i, M., \textit{On the norms of r-circulant matrices with the hyperharmonic numbers}, Journal of Mathematical Inequalities \textbf{10(2)}(2016), 445-458.
  • Bah\c{s}i, M., \textit{On the norms of circulant matrices with the generalized Fibonacci and Lucas numbers,} TWMS J. Pure Appl. Math. \textbf{6(1)}(2015), 84-92.
  • Bah\c{s}i, M., Mez\"{o}, I., Solak, S.,\textit{ A symmetric algorithm for hyper-Fibonacci and hyper-Lucas numbers}, Annales Mathematicae et Informaticae \textbf{43}(2014), 19--27.
  • Bah\c{s}i, M., S. Solak, \textit{On the norms of r-circulant matrices with the hyper-Fibonacci and Lucas numbers}, Journal of Mathematical Inequalities \textbf{8(4)}(2014), 693-705.
  • Bahsi, M., Solak, S., \textit{On the circulant matrices with arithmetic sequence}, Int. J. Cont. Math. Sciences \textbf{5(25)}(2010), 1213 -- 1222.
  • Cao, N-N., Zhao, F-Z., \textit{Some Properties of Hyperfibonacci and Hyperlucas Numbers,}\ Journal of Integer Sequences \textbf{13}(2010), Article 10.0.8.
  • Davis, P.J., \textit{Circulant Matrices,} Wiley, New York, Chichester, Brisbane, 1979.
  • Dil, A., Mez\"{o}, I., \textit{A symmetric algorithm for hyperharmonic and Fibonacci numbers,}\ Appl. Math. Comp. \textbf{206}(2008), 942-951.
  • Fischer, B., Modersitzki, J., \textit{Fast inversion of matrices arising in image processing,} Numer. Algorithms \textbf{22}(1999), 1--11.
  • Georgiou, S.D., Kravvaritis, C., \textit{New Good Quasi-Cyclic Codes over GF(3),} Int. J. Algebra \textbf{1(1)}(2007), 11--24.
  • Horn, R.A., Johnson, C.R., \textit{Matrix Analysis}, Cambridge University Press, Cambridge, 1985.
  • Horn, R.A., Johnson, C.R., \textit{Topics in Matrix Analysis}, Cambridge University Press, Cambridge, 1991.
  • Karner, H., Schneid, J., Ueberhuber, C.W.,\textit{Spectral Decomposition of Real Circulant Matrices,} Linear Algebra and Its Appl., \textbf{367}(2003), 301-311.
  • K\i z\i late\c{s}, C., Tuglu, N., \textit{On the bounds for the spectral norms of geometric circulant matrices}, Journal of Inequalities and Applications (2016), 2016:312.
  • Kocer, E.G., \textit{Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and Jacobsthal-Lucas numbers,} Hacettepe Journal of Mathematics and Statistics \textbf{36(2)}(2007), 133--142.
  • Kocer, E.G., Mansour, T., Tu\u{g}lu, N., \textit{Norms of circulant and semicirculant matrices with Horadam's numbers,} Ars Combinatoria \textbf{85}(2007), 353-359.
  • Tuglu, N., K\i z\i late\c{s}, C., \textit{On the norms of circulant and r-circulant matrices with the hyperharmonic Fibonacci numbers}, Journal of Inequalities and Applications (2015), 2015:253.
  • Tuglu, N., K\i z\i late\c{s}, C., \textit{On the Norms of Some Special Matrices with the Harmonic Fibonacci Numbers,} Gazi University Journal of Science \textbf{28(3)}(2015), 497-501.
  • \"{O}cal, A.A., Tu\u{g}lu, N., Alt\i n\i \c{s}\i k, E., \textit{On the representation of k-generalized Fibonacci and Lucas numbers}, Appl. Math. Comp. \textbf{170}(2005), 584-596.
  • Shen, S., Cen, J., \textit{On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers,} Appl. Math. Comp. \textbf{216}(2010), 2891-2897.
  • Solak, S., \textit{On the norms of circulant matrices with the Fibonacci and Lucas numbers}, Appl. Math. Comp. \textbf{160}(2005), 125-132.
  • Solak, S., \textit{Erratum to \textquotedblleft On the Norms of Circulant Matrices with the Fibonacci and Lucas Numbers\textquotedblright \ [Appl. Math. Comp., 160, (2005), 125-132],} Appl. Math. Comp. \textbf{190}(2007), 1855-1856.
  • T\"{u}rkmen, R., G\"{o}kba\c{s}, H., \textit{On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers}, Journal of Inequalities and Applications (2016),2016:65.
  • Yazlik, Y., Taskara, N., \textit{On the norms of an $r-$circulant matrix with the generalized k-Horadam numbers,} Journal of Inequalities and Applications (2013), 2013:394.
Toplam 25 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Mustafa Bahşi 0000-0002-6356-6592

Süleyman Solak 0000-0003-4085-277X

Yayımlanma Tarihi 31 Aralık 2020
Yayımlandığı Sayı Yıl 2020 Cilt: 12 Sayı: 2

Kaynak Göster

APA Bahşi, M., & Solak, S. (2020). On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers. Turkish Journal of Mathematics and Computer Science, 12(2), 76-85. https://doi.org/10.47000/tjmcs.715052
AMA Bahşi M, Solak S. On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers. TJMCS. Aralık 2020;12(2):76-85. doi:10.47000/tjmcs.715052
Chicago Bahşi, Mustafa, ve Süleyman Solak. “On The Norms of Another Form of $r-$Circulant Matrices With The Hyper-Fibonacci and Lucas Numbers”. Turkish Journal of Mathematics and Computer Science 12, sy. 2 (Aralık 2020): 76-85. https://doi.org/10.47000/tjmcs.715052.
EndNote Bahşi M, Solak S (01 Aralık 2020) On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers. Turkish Journal of Mathematics and Computer Science 12 2 76–85.
IEEE M. Bahşi ve S. Solak, “On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers”, TJMCS, c. 12, sy. 2, ss. 76–85, 2020, doi: 10.47000/tjmcs.715052.
ISNAD Bahşi, Mustafa - Solak, Süleyman. “On The Norms of Another Form of $r-$Circulant Matrices With The Hyper-Fibonacci and Lucas Numbers”. Turkish Journal of Mathematics and Computer Science 12/2 (Aralık 2020), 76-85. https://doi.org/10.47000/tjmcs.715052.
JAMA Bahşi M, Solak S. On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers. TJMCS. 2020;12:76–85.
MLA Bahşi, Mustafa ve Süleyman Solak. “On The Norms of Another Form of $r-$Circulant Matrices With The Hyper-Fibonacci and Lucas Numbers”. Turkish Journal of Mathematics and Computer Science, c. 12, sy. 2, 2020, ss. 76-85, doi:10.47000/tjmcs.715052.
Vancouver Bahşi M, Solak S. On The Norms of Another Form of $r-$Circulant Matrices with The Hyper-Fibonacci and Lucas Numbers. TJMCS. 2020;12(2):76-85.