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A new generalization of the Frank matrix and its some properties

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Abstract

Matrix theory is of great importance for the solution of some real-world problems and computational processes. Especially, some special type matrices and the linear algebraic properties of these matrices are considerably important for these computational processes. In this article, we propose a new r-Frank matrix, which is the general form of the Frank matrix, and other generalization of the Frank matrix. Then, we study its algebraic structure, some factorizations, determinant, inverse, and some norms. Moreover, we obtain miscellaneous linear algebra properties for the reciprocal matrix of the r-Frank matrix. Finally, to verify our results, as an interesting application, the mentioned above properties of the r -Frank matrix, whose entries are harmonic Fibonacci numbers, are presented.

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References

  • Akbiyik SY, Akbiyik M, Yilmaz F (2021) One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms. Mathematics. 9(5):539

    Article  Google Scholar 

  • Andrade E, Carrasco-Olivera D, Manzaneda C (2021) On circulant like matrices properties involving Horadam, Fibonacci, Jacobsthal and Pell numbers. Linear Algebra Appl 617:100–120

    Article  MathSciNet  Google Scholar 

  • Bertaccini D, Ng MK (2001) Skew-circulant preconditioners for systems of LMF-based ODE codes. Numer. Anal. Appl. LNCS. 93-101

  • Bhatia R (2006) Infinitely divisible matrices. Amer Math Monthly 113(2):221–235

    Article  MathSciNet  Google Scholar 

  • Bhatia R (2011) Min matrices and mean matrices. Math Intelligen 33(2):22–28

    Article  MathSciNet  Google Scholar 

  • da Fonseca CM (2007) On the eigenvalues of some tridiagonal matrices. J Comput Appl Math 200(1):283–286

    Article  MathSciNet  ADS  Google Scholar 

  • Deveci O (2016) The Pell-circulant sequences and their applications. Maejo Int J Sci Technol 10(3):284–293

    MathSciNet  Google Scholar 

  • Deveci O (2018) On The Fibonacci-Circulant \(p\)-Sequences. Util. Math. 108:107–124

    MathSciNet  Google Scholar 

  • Deveci O, Karaduman E, Campbell CM (2017) The Fibonacci-circulant sequences and their applications. Iran J Sci Technol Trans Sci 41(4):1033–1038

    Article  MathSciNet  Google Scholar 

  • Frank WL (1958) Computing eigenvalues of complex matrices by determinant evaluation and by methods of Danilewski and Wielandt. J Soc Ind Appl Math 6(4):378–392

    Article  MathSciNet  Google Scholar 

  • Hake J-F (1985) A remark on Frank matrices. Computing 35:375–379

    Article  MathSciNet  Google Scholar 

  • Horn RA (1990) The Hadamard product. Proc Sympos Appl Math 40:87–169

    Article  MathSciNet  Google Scholar 

  • Horn RA, Johnson CR (1991) Topics in matrix analysis. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Jiang ZL, Jiang ZW (2012) On the norms of RFPLR circulant matrices with the Fibonacci and Lucas numbers. SCET (Xi’an): 385-388

  • Jiang ZL, Xu ZB (2005) Efficient algorithm for finding the inverse and the group inverse of FLS \(r\)-circulant matrix. J Appl Math Comput 18:45–57

    Article  MathSciNet  Google Scholar 

  • Kılıc E, Arıkan T (2019) Studying new generalizations of Max-Min matrices with a novel approach. Turk J Math 43(4):2010–2024

    Article  MathSciNet  Google Scholar 

  • Kızılateş C, Terzioğlu N (2022) On \(r\)-min and \(r\) -max matrices. J Appl Math Comput 68:4559–4588

    Article  MathSciNet  Google Scholar 

  • Kızılateş C, Tuglu N (2016) On the bounds for the spectral norms of geometric circulant matrices. J. Inequal. Appl. 2016:312

    Article  MathSciNet  Google Scholar 

  • Kızılateş C, Tuglu N (2018) On the norms of geometric and symmetric geometric circulant matrices with the Tribonacci number. Gazi Univ. J. Sci. 31(2):555–567

    Google Scholar 

  • Kocer EG, Mansour T, Tuglu N (2007) Norms of circulant and semicirculant matrices with Horadam’s numbers. Ars Comb. 85:353–359

    MathSciNet  Google Scholar 

  • Koshy T (2019) Fibonacci and lucas numbers with applications. Wiley, New York, NY, USA

    Google Scholar 

  • Lyness JN, Sørevik T (2004) Four-dimensional lattice rules generated by skew-circulant matrices. Math Comput 73(245):279–295

    Article  MathSciNet  ADS  Google Scholar 

  • Mattila M, Haukkanen P (2016) Studying the various properties of min and max matrices-elementary vs. more advanced methods. Spec Matrix 4:101–109

    MathSciNet  Google Scholar 

  • Mersin EÖ (2023) Sturm’s theorem for Min matrices. AIMS Math 8(7):17229–17245

    Article  MathSciNet  Google Scholar 

  • Mersin EÖ, Bahsi M (2021) Sturm theorem for the generalized Frank matrix. Hacet J Math Stat 50(4):1002–1011

    Article  MathSciNet  Google Scholar 

  • Mersin EÖ, Bahsi M, Maden AD (2020) Some properties of generalized Frank matrices. Math Sci Appl E-Notes 8(2):170–177

    Article  Google Scholar 

  • Neudecker H, Trenkler G, Liu S (2009) Problem section. Stat Pap 50:221–223

    Article  Google Scholar 

  • Ozgul D, Bahsi M (2020) Min matrices with hyper Lucas numbers. J Sci Arts 4(53):855–864

    Article  Google Scholar 

  • Petroudi SHJ, Pirouz B (2015) A particular matrix, Its inversion and some norms. Appl Comput Math 4:47–52

    Article  Google Scholar 

  • Polatlı E (2019) On the bounds for the spectral norms of \(r\) -circulant matrices with a type of catalan triangle numbers. J Sci Art 3(48):575–578

    Google Scholar 

  • Pólya G, Szegő G (1925) Aufgaben und Lehrsätze aus der Analysis, Zweiter Band: Funktionentheorie, Nullstellen, Polynome, Determinanten, Zahlentheorie. Grundlagen Math. Wiss. Einzeld. 20, Springer, Berlin

  • Radicic B (2016) On \(k\)-circulant matrices (with geometric sequence). Quaest Math 39(1):135–144

    Article  MathSciNet  Google Scholar 

  • Radicic B (2017) On \(k\)-circulant matrices with arithmetic sequence. Filomat. 31(8):2517–2525

    Article  MathSciNet  Google Scholar 

  • Shen SQ, Cen JM (2010) On the bounds for the norms of \(r\) -circulant matrices with Fibonacci and Lucas numbers. Appl Math Comput 216:2891–2897

    MathSciNet  Google Scholar 

  • Shen N, Jiang ZL, Li J (2013) On explicit determinants of the RFMLR and RLMFL circulant matrices involving certain famous numbers. WSEAS Trans Math 12(1):42–53

    Google Scholar 

  • Shi BJ (2018) The spectral norms of geometric circulant matrices with the generalized \(k\)-Horadam numbers. J Inequal Appl 2018:14

    Article  MathSciNet  PubMed  PubMed Central  Google Scholar 

  • Shi BJ (2021) On the norms of RFMLR-circulant matrices with the exponential and trigonometric functions. J Math 2021:1–9

    MathSciNet  Google Scholar 

  • Solak S (2005) On the norms of circulant matrices with the Fibonacci and Lucas numbers. Appl Math Comput 160:125–132

    MathSciNet  Google Scholar 

  • Tuglu N, Kızılateş C (2015) On the norms of circulant and \(r\)-circulant matrices with the hyperharmonic Fibonacci numbers. J. Inequal. Appl. 2015:253

    Article  MathSciNet  Google Scholar 

  • Tuglu N, Kızılateş C, Kesim S (2015) On the harmonic and hyperharmonic Fibonacci numbers. Adv Differ Equ 2015:297

    Article  MathSciNet  Google Scholar 

  • Varch JM (1986) A generalization of the Frank matrix. SIAM J Sci Stat Comput 7(3):835–839

    Article  MathSciNet  Google Scholar 

  • Xu TT, Jiang ZL (2014) Explicit determinants of the RFP\(r\)LrR circulant and RLP\(r\)F\(r\)L circulant matrices involving some famous numbers. Abstr Appl Anal Article ID: 647030

  • Zhang F (1999) Matrix theory. Basic results and techniques. Springer, New York

    Book  Google Scholar 

  • Zhao G (2009) The improved nonsingularity on the \(r\) -circulant matrices in signal processing. Proc. Int. Conf. Comp. Tech. Deve. (Kota Kinabalu/Malaysia). 564-567

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Acknowledgements

Authors would like to thank Editors and the anonymous reviewers for their constructive comments in improving this paper. This work is supported by Scientific Research Program Funded by Shaanxi Provincial Education Department (Program No. 23JK0661).

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Authors and Affiliations

  1. School of Science, Xi’an University of Posts and Telecommunications, Xi’an, Shaanxi, People’s Republic of China

    Baijuan Shi

  2. Department of Mathematics, Zonguldak Bulent Ecevit University, Zonguldak, 67100, Turkey

    Can Kızılateş

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  1. Baijuan Shi
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  2. Can Kızılateş
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Correspondence to Baijuan Shi.

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Communicated by Jinyun Yuan.

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Shi, B., Kızılateş, C. A new generalization of the Frank matrix and its some properties. Comp. Appl. Math. 43, 19 (2024). https://doi.org/10.1007/s40314-023-02524-2

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  • DOI: https://doi.org/10.1007/s40314-023-02524-2

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