On k -circulant matrices involving the Jacobsthal numbers

• Autores: Biljana Radicic
• Localización: Revista de la Unión Matemática Argentina, ISSN 0041-6932, ISSN-e 1669-9637, Vol. 60, Nº. 2, 2019, págs. 431-442
• Idioma: inglés
• DOI: 10.33044/revuma.v60n2a10
• Enlaces
  • Texto completo (pdf)
• Resumen

  • Let k be a nonzero complex number. We consider a k-circulant matrix whose first row is (J1, J2, . . . , Jn), where Jn is the n th Jacobsthal number, and obtain the formulae for the eigenvalues of such matrix improving the formula which can be obtained from the result of Y. Yazlik and N. Taskara [J. Inequal. Appl. 2013, 2013:394, Theorem 7]. The obtained formulae for the eigenvalues of a k-circulant matrix involving the Jacobsthal numbers show that the result of Z. Jiang, J. Li, and N. Shen [WSEAS Trans. Math. 12 (2013), no. 3, 341–351, Theorem 10] is not always applicable. The Euclidean norm of such matrix is determined. We also consider a k-circulant matrix whose first row is (J −1 1 , J−1 2 , . . . , J−1 n ) and obtain the upper and lower bounds for its spectral norm

• Referencias bibliográficas
  • D. Bertaccini and M. K. Ng, Skew-circulant preconditioners for systems of LMF-based ODE codes. In: Numerical analysis and its applications...
  • D. Bozkurt and T-Y. Tam, Determinants and inverses of circulant matrices with Jacobsthal and Jacobsthal-Lucas numbers, Appl. Math. Comput....
  • D. Bozkurt and T-Y. Tam, Determinants and inverses of r-circulant matrices associated with a number sequence, Linear Multilinear Algebra 63...
  • R. E. Cline, R. J. Plemmons and G. Worm, Generalized inverses of certain Toeplitz matrices, Linear Algebra Appl. 8 (1974), no. 1, 25–33. MR...
  • R. L. Ellis and D. C. Lay, Factorization of finite rank Hankel and Toeplitz matrices, Linear Algebra Appl. 173 (1992), 19–38. MR 1170502.
  • M. C. Gouveia, Generalized invertibility of Hankel and Toeplitz matrices, Linear Algebra Appl. 193 (1993), 95–106. MR 1240274.
  • R. M. Gray, Toeplitz and circulant matrices: A review, Found. Trends Commun. Inf. Theory 2 (2006), no. 3, 155–239.
  • E. J. Hannan, Time Series Analysis, Methuen, London, 1960. MR 0114281.
  • C. He, J. Ma, K. Zhang and Z. Wang, The upper bound estimation on the spectral norm of r-circulant matrices with the Fibonacci and Lucas numbers,...
  • A. F. Horadam, Jacobsthal representation numbers, Fibonacci Quart. 34 (1996), no. 1, 40– 54. MR 1371475.
  • R. A. Horn, The Hadamard product. In: Matrix theory and applications (Phoenix, AZ, 1989), 87–169, Proc. Sympos. Appl. Math., 40, AMS Short...
  • R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge Univ. Press, Cambridge, 1991. MR 1091716.
  • I. S. Iohvidov, Hankel and Toeplitz Matrices and Forms: Algebraic Theory, Birkh¨auser, Boston, 1982. MR 0677503.
  • Z. Jiang, J. Li and N. Shen, On the explicit determinants and singularities of r-circulant and left r-circulant matrices with some famous...
  • E. Gokcen Kocer, Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and Jacobsthal-Lucas numbers, Hacet....
  • F. Koken and D. Bozkurt, On the Jacobsthal numbers by matrix methods, Int. J. Contemp. Math. Sci. 3 (2008), no. 13-16, 605–614. MR 2484951.
  • G. Labahn and T. Shalom, Inversion of Toeplitz matrices with only two standard equations, Linear Algebra Appl. 175 (1992), 143–158. MR 1179345.
  • S. Liu and G. Trenkler, Hadamard, Khatri-Rao, Kronecker and other matrix products, Int. J. Inf. Syst. Sci. 4 (2008), no. 1, 160–177. MR 2401772.
  • J. N. Lyness and T. Sørevik, Four-dimensional lattice rules generated by skew-circulant matrices, Math. Comput. 73 (2004), no. 245, 279–295....
  • M. K. Ng, Circulant and skew-circulant splitting methods for Toeplitz systems, J. Comput. Appl. Math. 159 (2003), no. 1, 101–108. MR 2022320.
  • B. Radiˇci´c, On k-circulant matrices (with geometric sequence), Quaest. Math. 39 (2016), no. 1, 135–144. MR 3483362.
  • B. Radiˇci´c, On k-circulant matrices with arithmetic sequence, Filomat 31 (2017), no. 8, 2517–2525. MR 3637047.
  • S. Q. Shen and J. M. Cen, On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput. 216...
  • W. Sintunavarat, The upper bound estimation for the spectral norm of r-circulant and symmetric r-circulant matrices with the Padovan sequence,...
  • W. F. Trench, An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Indust. Appl. Math. 12 (1964), no. 3, 515–522. MR 0173681.
  • Y. Yazlik and N. Taskara, On the norms of an r-circulant matrix with the generalized kHoradam numbers, J. Inequal. Appl. 2013, 2013:394, 8...
  • G. Zhao, The improved nonsingularity on the r-circulant matrices in signal processing. In: Proceedings, 2009 International Conference on Computer...
  • G. Zielke, Some remarks on matrix norms, condition numbers, and error estimates for linear equations, Linear Algebra Appl. 110 (1988), 29–41....