1918

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

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

PDF

___

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.
Bahc{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.
Bahc{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.
Bahc{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.
Bahc{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.
Ki zi latec{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., Tuu{g}lu, N., textit{Norms of circulant and semicirculant matrices with Horadam's numbers,} Ars Combinatoria textbf{85}(2007), 353-359.
Tuglu, N., Ki zi latec{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., Ki zi latec{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., Tuu{g}lu, N., Alti ni 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 Numberstextquotedblright [Appl. Math. Comp., 160, (2005), 125-132],} Appl. Math. Comp. textbf{190}(2007), 1855-1856.
T"{u}rkmen, R., G"{o}kbac{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.

Turkish Journal of Mathematics and Computer Science-Cover

ISSN: 2148-1830
Yayın Aralığı: Yılda 2 Sayı
Başlangıç: 2013
Yayıncı: MATEMATİKÇİLER DERNEĞİ

Arşiv