SONUÇ VE ÖNERİLER

Bu çalışmada bazı sayı dizilerine bağlı Circulant, Toeplitz, Hankel, Cauchy- Toeplitz ve Cauchy-Hankel yapısındaki matrisler tanımlanmıştır. Bu matrislerin spektral normları için alt ve üst sınırlar elde edilmiştir. Ancak Fermat, Mersenne, Catalan, Motzkin, Bell ve Happy gibi başka sayı dizilerine bağlı matrisler üzerinde çok durulamamıştır. Bu dizilere bağlı özel tipteki matrisler tanımlanarak normları ve diğer özelikleri incelenebilir.

Spektral norm değerleri ve elde edilen sınır değerleri sekizinci bölümde tablolar üzerinde kıyaslanmıştır. Tablolardan bazı alt ve üst sınırların iyi olmadığı görülmektedir. Daha iyi seçimler yapılarak veya varsa daha iyi eşitsizlikler kullanarak gerçek değere daha yakın sınırlar elde edilebilir.

KAYNAKLAR

[1] Alptekin E. G., Pell, Pell-Lucas ve Modified Pell sayıları ile tanımlı circulant ve semicirculant matrisler, Doktora Tezi, S. Ü. Fen Bilimleri Enstitüsü, Konya, 2005 [2] Altınışık E., Tuğlu N., Haukkanen P., A note on bounds for norms of the reciprocal LCM matrix, Mathematical Inequalities and Applications, 7(4), 491–496, 2004

[3] Belitski G. R., Lyubich Yu. I., Matrix Norms and Their Applications, Translated from the Russian by A. Iacob. Operator Theory: Advances and Applications, 36. Birkhuser Verlag, Basel, 1988

[4] Benoumhani M., A sequence of Binomial coefficients related to Lucas and Fibonacci numbers, Journal of Integer Sequences, Vol 6, Article 03.2.1, 2003

[5] Bozkurt D., On the bounds of the norms of almost Cauchy-Hankel matrices, Proceedings of the Int. Symposium on Mathematical and Computational Applications, 30–36, Manisa, 1996

p

[6] Bozkurt D., On the norms of Cauchy-Toeplitz matrices, Linear and Multilinear Algebra, 44, 341–346, 1998

p

[7] Bozkurt D., On the norms of Hadamard product of Cauchy-Toeplitz and Cauchy-Hankel matrices, Linear and Multilinear Algebra, 45, 333–339, 1999

p

[8] Caferov V., Anadolu Üniversitesi Açık Öğretim Fakültesi Yayınları No: 600, Eskişehir, 1999

[9] Cahill N. D., D’Errico J. R., Narayan D. A., Narayan J. Y., Fibonacci determinants, The College Math. Journal of The Math. Association of America, 33(3), 221–225, 2002

[10] Cerin Z., Sums of squares and products of Jacobsthal numbers, Journal. of Integer Sequences, Vol. 10, Article 07.2.5, 2007

[11] Civciv H., Türkmen R., Notes on norms of circulant matrices with Lucas number, International Journal of Information and Systems Sciences, 4(1), 142–147, 2008

[12] Civciv H., Türkmen R., On the bounds for the spectral and norms of the Khatri-Rao product of Cauchy-Hankel matrices, Journal of Inequalities in Pure and Applied Mathematics, 7(5), Article 195, 2006

p

[13] Davis P. J., Circulant Matrices, John Wiley and Sons, NY, 1979

[14] Er M. C., Sums of Fibonacci numbers by matrix methods, The Fibonacci Quarterly, 22(3), 204–207, 1984

[15] Gloub G. H., Loan C. F. V., Matrix Computations, The John Hopkins University Press, 1989

[16] Gray R. M., Toeplitz and Circulant Matrices: A Review, Now Publishers Inc, 2006

[17] Grenander U., Szegö G., Toeplitz Forms and Their Applications, University of California Press, Berkeley, 1958

[18] Güngör A. D., Türkmen R., Bounds for extreme singular values of a complex matrix and its applications, Mathematical Inequalities and Applications, 9(1), 23–31, 2006

[19] Güngör A. D., Lower bounds for the norms of Cauchy-Toeplitz and Cauchy- Hankel matrices, Applied Mathematics and Computation, 157, 599–604, 2004

[20] Haukkanen P., On the norm of GCD and related matrices, Journal of Inequalities in Pure and Applied Mathematics, 5(3), Article 61, 2004

p

[21] Horadam A. F., Applications of Modified Pell numbers to representations, Ulam Quarterly, 3(1), 34–53, 1994

[22] Horn R., Johnson C. R., Topics in Matrix Analysis, Cambridge University Press, 1991

[23] Karaduman E., An application of Fibonacci numbers in matrices, Applied Mathematics and Computation, 147, 903–908, 2004

[24] Karaduman E., On determinants of matrices with general Fibonacci numbers entries, Applied Mathematics and Computation, 167, 670–676, 2005

[25] Kayabaş H., Fibonacci dizilerinin uygulamaları, Yüksek Lisans Tezi, S. Ü. Fen Bilimleri Enstitüsü, Konya, 2006

k−

[26] Kocer E. G., Circulant, negacyclic and semicirculant matrices with the Modified Pell, Jacobsthal and Jacobsthal-Lucas numbers, Hacettepe Journal of Mathematics and Statistics, 36(2), 133–142, 2007

[27] Koshy T., Fibonacci and Lucas Numbers with Applications, John Wiley and Sons, NY, 2001

[28] Kölbig K.S., Complex zeros of two incomplete Riemann zeta functions, Math. Comp. 26, 551–565, 1972

[29] Lee G. Y., Kim J. S., Cho S. H., Some combinatorial identities via Fibonacci numbers, Discrete Applied Mathematics, 130, 527–534, 2003

[30] Lee G. Y., Kim J. S., The linear algebra of the k-Fibonacci matrix, Linear Algebra and its Applications, 373, 75–87, 2003

[31] Mathias R., The spectral norm of a nonnegative matrix. Linear Algebra and its Applications, 131, 269–284, 1990

[32] Melham R., Sums involving Fibonacci and Pell numbers, Portugaliae Mathematica, Vol. 56, Fasc. 3, 1999

[33] Merikoski J. K., Kumar R., Lower bounds for the spectral norm, Journal of Inequalities in Pure and Applied Mathematics, 6(4), Article 124, 2005

[34] Moenck R., On computing closed forms for summations, in: Proceedings of the MACSYMA User’s Conference, 225–236, 1977

[35] Noble B., Applied Linear Algebra, Englewood Cliffs, N. J., Prentince-Hall, 1969

[36] Öcal A. A., Tuğlu N., Altınışık E., On the representation of k-generalized Fibonacci and Lucas numbers, Applied Mathematics and Computation, 170, 584– 596, 2005

[37] Parter S. V., On the distribution of the singular values of Toeplitz matrices, Linear Algebra and its Applications, 80, 115–130, 1986

[38] Roch S., Silbermann B., A note on singular values of Cauchy-Toeplitz matrices, Linear Algebra and its Applications, Vol. 275–276, 531–536, 1998

[39] Solak S., Bozkurt D., A note on bounds for the norms of Cauchy-Hankel Matrices, Numerical Linear Algebra with Applications, 10(4), 377–382, 2003

[40] Solak S., Bozkurt D., On the spectral norms of Cauchy-Toeplitz and Cauchy- Hankel matrices, Applied Mathematics and Computation, 140, 231–238, 2003

[41] Solak S., Türkmen R., Bozkurt D., On the norms of GCD, Toeplitz and Hankel matrices related to Fibonacci numbers, International Mathematical Journal, 3(2), 195–200, 2003

[42] Solak S., Türkmen R., Bozkurt D., On GCD, LCM and Hilbert matrices and their applications”, Applied Mathematics and Computation, 146, 595–600, 2003 [43] Solak S., On the norms of circulant matrices with the Fibonacci and Lucas numbers, Applied Mathematics and Computation, 160, 125–132, 2005

[44] Taşçı D., Kılıç E., On the order-k generalized Lucas numbers, Applied Mathematics and Computation, 155, 637–641, 2004

[45] Taşçı D., Lineer Cebir, Sel-Ün Vakfı Yayınları, No: 9, Konya, 2001

[46] Türkmen R., Bozkurt D., On the bounds for the norms of Cauchy-Toeplitz and Cauchy-Hankel matrices, Applied Mathematics and Computation, 132, 633–642, 2002

[47] Türkmen R., Bozkurt D., A note for bounds of norms of Hadamard product of matrices, Mathematical Inequalities and Applications, 9(2), 211–217, 2006

[48] Tyrtyshnikov E. E., Cauchy-Toeplitz matrices and some applications, Linear Algebra and its Applications, 149, 1 – 18, 1991

[49] Vajda S., Fibonacci and Lucas Numbers, and the Golden Section, John Wiley and Sons, NY, 1989

[50] Wu H., On the lower bounds for the norms of Cauchy–Toeplitz and Cauchy– Hankel matrices, Applied Mathematics and Computation 205, 391–395, 2008

[51] Yalçıner A., Taşkara N., On the norms of Hankel matrices with the Motzkin numbers, Selçuk Journal of Applied Mathematics, 8(1), 9–14, 2007

[52] Yalçıner A., spectral norms of some special circulant matrices, Int. J. Contemp. Math. Sciences, 3(35), 1733–1738, 2008

[53] Zhang F., Matrix Theory, Basic Results and Techniques, Springer-Verlag, NY, 1999