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Year 2019, Volume: 48 Issue: 3, 805 - 817, 15.06.2019

Abstract

References

  • E. Boman, The Moore-Penrose Pseudoinverse of an Arbitrary, Square, k-circulant Matrix, Linear Multilinear Algebra 50 (2), 175-179, 2002.
  • A.C.F. Bueno, Right Circulant Matrices With Geometric Progression, Int. J. Appl. Math. Res. 1 (4), 593-603, 2012.
  • A.C.F. Bueno, Generalized Right Circulant Matrices with Geometric Sequence, Int. J. Math. Sci. Comput. 3 (1), 17-18, 2013.
  • A.C.F. Bueno, Right circulant matrices with ratio of the elements of Fibonacci and geometric sequence, Notes Numb. Theory Discr. Math. 22 (3), 79-83, 2016.
  • R.E. Cline, R.J. Plemmons and G. Worm, Generalized Inverses of Certain Toeplitz Matrices, Linear Algebra Appl. 8 (1), 25-33, 1974.
  • R. A. Horn, The Hadamard product, Proc. Sympos. Appl. Math. 40, 87-169, 1990.
  • R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
  • Z. Jiang, H. Xin and H.Wang, On computing of positive integer powers for r-circulant matrices, Appl. Math. Comput. 265, 409-413, 2015.
  • Z. Jiang and J. Zhou, A note on spectral norms of even-order r-circulant matrices, Appl. Math. Comput. 250, 368-371, 2015.
  • S. Liu and G. Trenkler, Hadamard, Khatri-Rao, Kronecker and other matrix products, Int. J. Inf. Syst. Sci. 4 (1), 160-177, 2008.
  • B. Radičić, On k-circulant matrices (with geometric sequence), Quaest. Math. 39 (1), 135-144, 2016.
  • Y. Yazlik and N. Taskara, Spectral norm, Eigenvalues and Determinant of Circulant Matrix involving the Generalized k-Horadam numbers, Ars Combin. 104, 505-512, 2012.
  • Y. Yazlik and N. Taskara, On the inverse of circulant matrix via generalized k- Horadam numbers, Appl. Math. Comput. 223, 191-196, 2013.
  • J. Zhou and Z. Jiang, Spectral norms of circulant-type matrices involving some well- known numbers, WSEAS Trans. Math. 12 (12), 1184-1194, 2013.
  • J. Zhou and Z. Jiang, Spectral norms of circulant-type matrices with Binomial Coef- ficients and Harmonic numbers, Int. J. Comput. Methods 11 (5), 2014.
  • J. Zhou and Z. Jiang, Spectral Norms of Circulant and Skew-Circulant Matrices with Binomial Coefficients Entries, Proc. 9th Inter. Symp. Lin. Drives Ind. Appl. 2, 219- 224, 2014.
  • G. Zielke, Some remarks on matrix norms, condition numbers, and error estimates for linear equations, Linear Algebra Appl. 110, 29-41, 1988.

On $k$-circulant matrices involving geometric sequence

Year 2019, Volume: 48 Issue: 3, 805 - 817, 15.06.2019

Abstract

In this paper we consider a $k$-circulant matrix with geometric sequence, where $k$ is a nonzero complex number. The eigenvalues, the determinant, the Euclidean norm and bounds for the spectral norm of such matrix are investigated. The method for obtaining the inverse of a nonsingular $k$-circulant matrix, was presented in [On $k$-circulant matrices (with geometric sequence), Quaest. Math. 2016]. A generalization of that method is given in this paper, and using it, the inverse of a nonsingular $k$-circulant matrix with geometric sequence is obtained. The Moore-Penrose inverse of a singular $k$-circulant matrix with geometric sequence is determined in a different way than the way using in [On $k$-circulant matrices (with geometric sequence), Quaest. Math. 2016].

References

  • E. Boman, The Moore-Penrose Pseudoinverse of an Arbitrary, Square, k-circulant Matrix, Linear Multilinear Algebra 50 (2), 175-179, 2002.
  • A.C.F. Bueno, Right Circulant Matrices With Geometric Progression, Int. J. Appl. Math. Res. 1 (4), 593-603, 2012.
  • A.C.F. Bueno, Generalized Right Circulant Matrices with Geometric Sequence, Int. J. Math. Sci. Comput. 3 (1), 17-18, 2013.
  • A.C.F. Bueno, Right circulant matrices with ratio of the elements of Fibonacci and geometric sequence, Notes Numb. Theory Discr. Math. 22 (3), 79-83, 2016.
  • R.E. Cline, R.J. Plemmons and G. Worm, Generalized Inverses of Certain Toeplitz Matrices, Linear Algebra Appl. 8 (1), 25-33, 1974.
  • R. A. Horn, The Hadamard product, Proc. Sympos. Appl. Math. 40, 87-169, 1990.
  • R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1991.
  • Z. Jiang, H. Xin and H.Wang, On computing of positive integer powers for r-circulant matrices, Appl. Math. Comput. 265, 409-413, 2015.
  • Z. Jiang and J. Zhou, A note on spectral norms of even-order r-circulant matrices, Appl. Math. Comput. 250, 368-371, 2015.
  • S. Liu and G. Trenkler, Hadamard, Khatri-Rao, Kronecker and other matrix products, Int. J. Inf. Syst. Sci. 4 (1), 160-177, 2008.
  • B. Radičić, On k-circulant matrices (with geometric sequence), Quaest. Math. 39 (1), 135-144, 2016.
  • Y. Yazlik and N. Taskara, Spectral norm, Eigenvalues and Determinant of Circulant Matrix involving the Generalized k-Horadam numbers, Ars Combin. 104, 505-512, 2012.
  • Y. Yazlik and N. Taskara, On the inverse of circulant matrix via generalized k- Horadam numbers, Appl. Math. Comput. 223, 191-196, 2013.
  • J. Zhou and Z. Jiang, Spectral norms of circulant-type matrices involving some well- known numbers, WSEAS Trans. Math. 12 (12), 1184-1194, 2013.
  • J. Zhou and Z. Jiang, Spectral norms of circulant-type matrices with Binomial Coef- ficients and Harmonic numbers, Int. J. Comput. Methods 11 (5), 2014.
  • J. Zhou and Z. Jiang, Spectral Norms of Circulant and Skew-Circulant Matrices with Binomial Coefficients Entries, Proc. 9th Inter. Symp. Lin. Drives Ind. Appl. 2, 219- 224, 2014.
  • G. Zielke, Some remarks on matrix norms, condition numbers, and error estimates for linear equations, Linear Algebra Appl. 110, 29-41, 1988.
There are 17 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Biljana Radičić This is me 0000-0002-5845-8503

Publication Date June 15, 2019
Published in Issue Year 2019 Volume: 48 Issue: 3

Cite

APA Radičić, B. (2019). On $k$-circulant matrices involving geometric sequence. Hacettepe Journal of Mathematics and Statistics, 48(3), 805-817.
AMA Radičić B. On $k$-circulant matrices involving geometric sequence. Hacettepe Journal of Mathematics and Statistics. June 2019;48(3):805-817.
Chicago Radičić, Biljana. “On $k$-Circulant Matrices Involving Geometric Sequence”. Hacettepe Journal of Mathematics and Statistics 48, no. 3 (June 2019): 805-17.
EndNote Radičić B (June 1, 2019) On $k$-circulant matrices involving geometric sequence. Hacettepe Journal of Mathematics and Statistics 48 3 805–817.
IEEE B. Radičić, “On $k$-circulant matrices involving geometric sequence”, Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 3, pp. 805–817, 2019.
ISNAD Radičić, Biljana. “On $k$-Circulant Matrices Involving Geometric Sequence”. Hacettepe Journal of Mathematics and Statistics 48/3 (June 2019), 805-817.
JAMA Radičić B. On $k$-circulant matrices involving geometric sequence. Hacettepe Journal of Mathematics and Statistics. 2019;48:805–817.
MLA Radičić, Biljana. “On $k$-Circulant Matrices Involving Geometric Sequence”. Hacettepe Journal of Mathematics and Statistics, vol. 48, no. 3, 2019, pp. 805-17.
Vancouver Radičić B. On $k$-circulant matrices involving geometric sequence. Hacettepe Journal of Mathematics and Statistics. 2019;48(3):805-17.