Authors:
Carlos M. da Fonseca Kuwait College of Science and Technology, Doha District, Block 4, P.O. Box 27235, Safat 13133, Kuwait University of Primorska, FAMNIT, Glagoljsaška 8, 6000 Koper, Slovenia

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Victor Kowalenko School of Mathematics and Statistics, University of Melbourne, Victoria 3010, Australia

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László Losonczi Faculty of Economics, University of Debrecen, Hungary

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Abstract

This survey revisits Jenő Egerváry and Otto Szász’s article of 1928 on trigonometric polynomials and simple structured matrices focussing mainly on the latter topic. In particular, we concentrate on the spectral theory for the first type of the matrices introduced in the article, which are today referred to as k-tridiagonal matrices, and then discuss the explosion of interest in them over the last two decades, most of which could have benefitted from the seminal article, had it not been overlooked.

  • [1]

    Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, ninth edition, Dover Publications, Inc., New York, 1970.

    • Search Google Scholar
    • Export Citation
  • [2]

    Bebiano, N. and Furtado, S., A reducing approach for symmetrically sparse banded and anti-banded matrices, Linear Algebra Appl., 581 (2019), 3650.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [3]

    Berman, A. and Hershkowitz, D., Characterization of acyclic d-stable matrices, Linear Algebra Appl., 58 (1984), 1731.

  • [4]

    Berman, A. and Hershkowitz, D., Matrix diagonal stability and its applications, SIAM J. Algebr. Discrete Methods, 4 (1983), 377382.

  • [5]

    Buschman, R. G., Fibonacci numbers, Chebyshev polynomials generalizations and difference equations, Fibonacci Quart., 1 (1963), 18, 19.

    • Search Google Scholar
    • Export Citation
  • [6]

    Duru, H. K. and Bozkurt, D., Integer powers of certain complex pentadiagonal Toeplitz matrices, Applied and Computational Matrix Analysis, Springer Proceedings in Mathematics & Statistics, 192, 199218, 2017.

    • Search Google Scholar
    • Export Citation
  • [7]

    Egerváry, E. and Szász, O., Einige Extremalprobleme im Bereiche der trigono- metrischen Polynome, Math. Z, 27 (1928), 641652.

  • [8]

    Ekström, S.-E. and Serra-Capizzano, S.., Eigenvalues and eigenvectors of banded Toeplitz matrices and the related symbols, Numer. Linear Algebra Appl., 25 (2018), e2137.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [9]

    Elsner, L. and Redheffer, R. M., Remarks on band matrices, Numer. Math., 10 (1967), 153161.

  • [10]

    Fejér, L., Über trigonometrische Polynome, J. Reine Angew. Math., 146 (1916), 5382.

  • [11]

    Da Fonseca, C. M., Eigenpairs of some particular band Toeplitz matrices: A comment, Numer. Linear Algebra Appl., 27 (2020), e2270

  • [12]

    Da Fonseca, C. M., On some conjectures regarding tridiagonal matrices, J. Appl. Math. Comput. Mech., 17 (2018), 1317.

  • [13]

    Da Fonseca, C. M., An identity between the determinant and the permanent of Hessenberg type-matrices, Czechoslovak Math. J., 61(136) (2011), 917921.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [14]

    Da Fonseca, C. M. and Kowalenko, V., Eigenpairs of a family of tridiagonal matrices: three decades later, Acta Math. Hungar., 160 (2020), 376389.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [15]

    Da Fonseca, C. M. and Petronilho, J., Explicit inverse of a tridiagonal k-Toeplitz matrix, Numer. Math., 100 (2005), no.3, 457482.

  • [16]

    Da Fonseca, C. M. and Petronilho, J., Explicit inverses of some tridiagonal matrices, Linear Algebra Appl., 325 (2001), 721.

  • [17]

    DA FONSECA, C. M. YILMAZ, F., Some comments on k-tridiagonal matrices: determinant, spectra, and inversion, Appl. Math. Comput., 270 (2015), 644647.

    • Search Google Scholar
    • Export Citation
  • [18]

    Gutiérrez-Gutiérrez, J., Singular value decomposition for comb filter matrices, Appl. Math. Comput., 222 (2013), 472477.

  • [19]

    Hadj, D. A. and Elouafi, M., A fast numerical algorithm for the inverse of a tridiagonal and pentadiagonal matrix, Appl. Math. Comput., 202 (2008), 441445.

    • Search Google Scholar
    • Export Citation
  • [20]

    Han, G.-N. and Krattenthaler, C., Rectangular Scott-type permanents, Séminaire Lotharingien Combin., 43 (2000), Article B43g, 25 pp.

  • [21]

    Horadam, A. F., Basic properties of a certain generalized sequence of numbers, Fibonacci Quart., 3 (1965), 161176.

  • [22]

    Jia, J. and Li, S., Symbolic algorithms for the inverses of general k-tridiagonal matrices, Comput. Math. Appl., 70 (2015), 30323042.

  • [23]

    Jia, J., Sogabe, T. and El-Mikkawy, M., Inversion of k-tridiagonal matrices with Toeplitz structure, Comput. Math. Appl., 65 (2013), 116125.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [24]

    KILIC, E., On a constant-diagonals matrix, Appl. Math. Comput., 204 (2008), 184190.

  • [25]

    Kirklar, E. and Yilmaz, F., A note on k-tridiagonal k-Toeplitz matrices, Alabama J. Math., 39 (2015).

  • [26]

    Kouachi, S., Explicit eigenvalues of some perturbed heptadiagonal matrices via recurrent sequences, Lobachevskii J. Math., 36 (2015), 2837.

  • [27]

    Krattenthaler, C., Advanced determinant calculus: A complement, Linear Al gebra Appl., 411 (2005), 68166.

  • [28]

    KÜÇÜk, A. Z. and DÜz, M., Relationships between the permanents of a certain type of k-tridiagonal symmetric Toeplitz matrix and the Chebyshev polynomials, J. Appl. Math. Comput. Mech., 16 (2017), 7586.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [29]

    KÜÇÜk, A. Z., Özen, M. INCE, H., Recursive and combinational formulas for permanents of general k-tridiagonal Toeplitz matrices, Filomat, 33 (2019), 307317.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [30]

    Lauret, E. A., The smallest Laplace eigenvalue of homogeneous 3-spheres, Bull. London Math. Soc., 51 (2019), 49-69.

  • [31]

    Lin, Y. and Lin, X., A novel algorithm for inverting a k-pentadiagonal matrix, The 2016 3rd International Conference on Systems and Informatics (ICSAI 2016), 578582.

    • Search Google Scholar
    • Export Citation
  • [32]

    Losonczi, L., Eigenvalues and eigenvectors of some tridiagonal matrices, Acta Math. Hung., 60 (1992), 309332.

  • [33]

    Losonczi, L., On some discrete quadratic inequalities, Int. Ser. Numer. Math., 80 (1987), 7385.

  • [34]

    McMillen, T., On the eigenvalues of double band matrices, Linear Algebra Appl., 431 (2009), 18901897.

  • [35]

    El-Mikkawy, M., A generalized symbolic Thomas algorithm, Appl. Math., 3 (2012), 342345.

  • [36]

    El-Mikkawy, M. and Atlan, F., A fast and reliable algorithm for evaluating n-th order k-tridiagonal determinants, Malaysian J. Math. Sci., 3 (2015), 349365.

    • Search Google Scholar
    • Export Citation
  • [37]

    El-Mikkawy, M. and Atlan, F., A new recursive algorithm for inverting general k-tridiagonal matrices, Appl. Math. Lett., 44 (2015), 3439.

  • [38]

    El-Mikkawy, M. and Atlan, F., A novel algorithm for inverting a general k- tridiagonal matrix, Appl. Math. Lett., 32 (2014), 4147.

  • [39]

    El-Mikkawy, M. and Sogabe, T., A new family of k-Fibonacci numbers, Appl. Math. Comput., 215 (2010), 44564461.

  • [40]

    OHASHI, A., SoGBE, T., and Usuda, T. S., On decomposition of k-tridiagonal l-Toeplitz matrices and its applications, Spec. Matrices, 3 (2015), 200206.

    • Search Google Scholar
    • Export Citation
  • [41]

    OKAYASU, T. and Ueta, Y., Estimates for moduli of coefficients of positive trigono- metric polynomials, Sci. Math. Jpn., 56 (2002), 115122.

    • Search Google Scholar
    • Export Citation
  • [42]

    Parter, S. V. and Youngs, J. W. T., The symmetrization of matrices by diagonal matrices, J. Math. Anal. Appl., 4 (1962), 102110.

  • [43]

    Popescu, G., Bohr inequalities for free holomorphic functions on polyballs, Adv. Math., 347 (2019), 10021053.

  • [44]

    Rimas, J., On computing of arbitrary positive integer powers for one type of sym metric pentadiagonal matrices of odd order, Appl. Math. Comput., 204 (2008), 120129.

    • Search Google Scholar
    • Export Citation
  • [45]

    Rimas, J., On computing of arbitrary positive integer powers for one type of symmetric pentadiagonal matrices of even order, Appl. Math. Comput., 203 (2008), 582591.

    • Search Google Scholar
    • Export Citation
  • [46]

    Rózsa, P., On periodic continuants, Linear Algebra Appl., 2 (1969) 267274.

  • [47]

    El-Shehawey, M. and El-Shreef, Gh. A., On a Markov chain roulette-type game, J. Phys. A.: Math. Theor., 42 (2009), 195005.

  • [48]

    SoGBE T., and El-Mikkawy, M., Fast block diagonalization of k-tridiagonal matrices, Appl. Math. Comput., 218 (2011), 27402743.

  • [49]

    SoGBE T., and Yilmaz, F., A note on a fast breakdown-free algorithm for computing the determinants and the permanents of fc-tridiagonal matrices, Appl. Math. Comput., 249 (2014), 98102.

    • Search Google Scholar
    • Export Citation
  • [50]

    Takahira, S., Sogbe T., and Usuda T. S., Bidiagonalization of (k,k + 1)- tridiagonal matrices, Spec. Matrices, 7 (2019), 2026.

  • [51]

    TĂnĂsescu, A. and Popescu, P. G., A fast singular value decomposition algorithm of general k-tridiagonal matrices, J. Comput. Sci., 31 (2019), 15.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • [52]

    Weisstein, E. W., Permutation Matrix. From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PermutationMatrix.html

  • [53]

    WituLa, R. and SLota, D., On computing the determinants and inverses of some special type of tridiagonal and constant-diagonals matrices, Appl. Math. Comput, 189 (2007), 514527.

    • Search Google Scholar
    • Export Citation
  • [54]

    YalÇiner, A., The LU factorizations and determinants of the k-tridiagonal matri ces, Asian-Eur. J. Math., 4 (2011), 187197.

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Studia Scientiarum Mathematicarum Hungarica
Language English
French
German
Size B5
Year of
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1966
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1
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4
Founder Magyar Tudományos Akadémia  
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ISSN 0081-6906 (Print)
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