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Year 2022, Volume: 10 Issue: 1, 188 - 196, 15.04.2022

Seyyed Hossein Jafari Petroudi Maryam Pirouz Mücahit Akbıyık Fatih Yılmaz

Abstract

References

  • [1] M. Akbulak, A. Ipek, Hadamard exponential Hankel Matrix, its eigenvalues and some norms, Math. sci. Lett, 1 (2012), 81-87.
  • [2] R. Reams, Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra and its Applications, 288 (1999), 35-43.
  • [3] M. Bahsi, S. Solak, on the matrices with Harmonic numbers, GU. J. Sc, 23(4) (2010), 445-448.
  • [4] M. Bahsi, on the norms of r−circulant matrices wit the hyper Harmonic numbers, Journal of mathematical inequalities, 10(2) (2016), 445-458.
  • [5] M. Istvan, D. Ayhan, Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence Cent. Eur. J. Math, 10(2) (2009), 1-12.
  • [6] D.Bozkurt, A note on the spectral norms of the matrices connected integer numbers sequence, Math.GM,. 1724v1 (2011), 171-190.
  • [7] H. Civciv, R. Turkmen, On the bounds for the spectral and ℓp norms of the Khatri-Rao products of Cauchy-Hankel matrices . Jipam Vol.7 Article 195 (2006), 365-380.
  • [8] E. Dupree, B.Mathes, Singular values of k-Fibonacci and k-Lucas Hankel matrix, Int. J. Contemp. Math. Science. Vol 7 (2012), no. 47, 2327–2339.
  • [9] A. D. Godase, M. B. Dhakne, On the properties of generalized multiplicative coupled Fibonacci sequence of r-th order, Int. J. Adv. Appl. Math. andMech. 2(3) (2015), 252 - 257.
  • [10] A. D. Godase, M. B. Dhakne, On the properties of generalized Fibonacci like polynomials, Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 234 - 251.
  • [11] A. D. Godase, M. B. Dhakne, On the properties of k-Fibonacci and k-Lucas numbers, Int. J. Adv. Appl. Math. andMech. 2(1) (2014), 100 - 106.
  • [12] S.H.J. Petroudi, B. Pirouz, On some properties of (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence, Int. J.Adv. Appl. Math. and Mech. 3(1) (2015) 98-101.
  • [13] S.H.J. Petroudi, B. Pirouz, A particular matrix, Its inversion and some norms, Appl. and Computational Math. 4(2) (2015), 47-52.
  • [14] A. Nalli, M.Sen, On the norms of circulant matrices with generalized Fibonacci numbers, Selc¸uk J. Appl. Math, Vol.11, No.1 (2010), 107-116.
  • [15] S. Solak, Bahsi. M, A particular matrix and its some properties, Scientific Research and Essays, Vol.8(1), (2013), 1-5.
  • [16] S. Solak, M. Bahsi, On the spectral norms of Toeplitz matrices with Fibonacci and Lucas numbers, Nonlinear Analysis, Hacettepe Journal of Mathematics and Statistics. 42,(2013), no. 1, 15-19.
  • [17] N. Tuglu, C. Kizilates, S. Kesim , On the harmonic and hyperharmonic Fibonacci numbers , Advances in Difference Equations, (2015), 1-12.
  • [18] F.Zhang, Matrix Theory, Basic results and techniques, Springer, 2011.
  • [19] S.H.J. Petroudi, M. Pirouz, Toward Special Symmetric Matrices with Harmonic Numbers, 8th National Conference on Mathematics of Payame Noor University, 2016.
  • [20] S. Yamac¸ Akbıyık, M. Akbıyık, F. Yılmaz, One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms. Mathematics 9(2021), 539.
  • [21] A. Dagdeviren, F. Kuruz, Special Real and Dual Matrices with Hadamard Product, Journal of Engineering Technology and Applied Sciences, 6(2) (2021), 127-134.

Some Special Matrices with Harmonic Numbers

Year 2022, Volume: 10 Issue: 1, 188 - 196, 15.04.2022

Seyyed Hossein Jafari Petroudi Maryam Pirouz Mücahit Akbıyık Fatih Yılmaz

Abstract

In this paper, we define a particular $n\times n$ matrix $H=[H_{k_{i,j}}]_{i,j=1}^{n}$ and its Hadamard exponential matrix $e^{\circ H}=[e^{H_{k_{i,j}}}]$, where $k_{i,j}=min(i,j)$ and $H_n$ is the $n^{th}$ harmonic number. Determinants and inverses of these matrices are investigated. Moreover, the Euclidean norm and two upper bounds and lower bounds for the spectral norm of these matrices are presented. Finally, we derive some identities about principal minors of these matrices.

Keywords

Harmonic number Spectral norm Hadamard inverse determinant

References

  • [1] M. Akbulak, A. Ipek, Hadamard exponential Hankel Matrix, its eigenvalues and some norms, Math. sci. Lett, 1 (2012), 81-87.
  • [2] R. Reams, Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra and its Applications, 288 (1999), 35-43.
  • [3] M. Bahsi, S. Solak, on the matrices with Harmonic numbers, GU. J. Sc, 23(4) (2010), 445-448.
  • [4] M. Bahsi, on the norms of r−circulant matrices wit the hyper Harmonic numbers, Journal of mathematical inequalities, 10(2) (2016), 445-458.
  • [5] M. Istvan, D. Ayhan, Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence Cent. Eur. J. Math, 10(2) (2009), 1-12.
  • [6] D.Bozkurt, A note on the spectral norms of the matrices connected integer numbers sequence, Math.GM,. 1724v1 (2011), 171-190.
  • [7] H. Civciv, R. Turkmen, On the bounds for the spectral and ℓp norms of the Khatri-Rao products of Cauchy-Hankel matrices . Jipam Vol.7 Article 195 (2006), 365-380.
  • [8] E. Dupree, B.Mathes, Singular values of k-Fibonacci and k-Lucas Hankel matrix, Int. J. Contemp. Math. Science. Vol 7 (2012), no. 47, 2327–2339.
  • [9] A. D. Godase, M. B. Dhakne, On the properties of generalized multiplicative coupled Fibonacci sequence of r-th order, Int. J. Adv. Appl. Math. andMech. 2(3) (2015), 252 - 257.
  • [10] A. D. Godase, M. B. Dhakne, On the properties of generalized Fibonacci like polynomials, Int. J. Adv. Appl. Math. and Mech. 2(3) (2015) 234 - 251.
  • [11] A. D. Godase, M. B. Dhakne, On the properties of k-Fibonacci and k-Lucas numbers, Int. J. Adv. Appl. Math. andMech. 2(1) (2014), 100 - 106.
  • [12] S.H.J. Petroudi, B. Pirouz, On some properties of (k,h)-Pell sequence and (k,h)-Pell-Lucas sequence, Int. J.Adv. Appl. Math. and Mech. 3(1) (2015) 98-101.
  • [13] S.H.J. Petroudi, B. Pirouz, A particular matrix, Its inversion and some norms, Appl. and Computational Math. 4(2) (2015), 47-52.
  • [14] A. Nalli, M.Sen, On the norms of circulant matrices with generalized Fibonacci numbers, Selc¸uk J. Appl. Math, Vol.11, No.1 (2010), 107-116.
  • [15] S. Solak, Bahsi. M, A particular matrix and its some properties, Scientific Research and Essays, Vol.8(1), (2013), 1-5.
  • [16] S. Solak, M. Bahsi, On the spectral norms of Toeplitz matrices with Fibonacci and Lucas numbers, Nonlinear Analysis, Hacettepe Journal of Mathematics and Statistics. 42,(2013), no. 1, 15-19.
  • [17] N. Tuglu, C. Kizilates, S. Kesim , On the harmonic and hyperharmonic Fibonacci numbers , Advances in Difference Equations, (2015), 1-12.
  • [18] F.Zhang, Matrix Theory, Basic results and techniques, Springer, 2011.
  • [19] S.H.J. Petroudi, M. Pirouz, Toward Special Symmetric Matrices with Harmonic Numbers, 8th National Conference on Mathematics of Payame Noor University, 2016.
  • [20] S. Yamac¸ Akbıyık, M. Akbıyık, F. Yılmaz, One Type of Symmetric Matrix with Harmonic Pell Entries, Its Inversion, Permanents and Some Norms. Mathematics 9(2021), 539.
  • [21] A. Dagdeviren, F. Kuruz, Special Real and Dual Matrices with Hadamard Product, Journal of Engineering Technology and Applied Sciences, 6(2) (2021), 127-134.
There are 21 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Seyyed Hossein Jafari Petroudi 0000-0003-4127-9215

Maryam Pirouz

Mücahit Akbıyık 0000-0002-0256-1472

Fatih Yılmaz

Publication Date April 15, 2022
Submission Date February 7, 2021
Acceptance Date December 8, 2021
Published in Issue Year 2022 Volume: 10 Issue: 1

Cite

APA Jafari Petroudi, S. H., Pirouz, M., Akbıyık, M., Yılmaz, F. (2022). Some Special Matrices with Harmonic Numbers. Konuralp Journal of Mathematics, 10(1), 188-196.
AMA Jafari Petroudi SH, Pirouz M, Akbıyık M, Yılmaz F. Some Special Matrices with Harmonic Numbers. Konuralp J. Math. April 2022;10(1):188-196.
Chicago Jafari Petroudi, Seyyed Hossein, Maryam Pirouz, Mücahit Akbıyık, and Fatih Yılmaz. “Some Special Matrices With Harmonic Numbers”. Konuralp Journal of Mathematics 10, no. 1 (April 2022): 188-96.
EndNote Jafari Petroudi SH, Pirouz M, Akbıyık M, Yılmaz F (April 1, 2022) Some Special Matrices with Harmonic Numbers. Konuralp Journal of Mathematics 10 1 188–196.
IEEE S. H. Jafari Petroudi, M. Pirouz, M. Akbıyık, and F. Yılmaz, “Some Special Matrices with Harmonic Numbers”, Konuralp J. Math., vol. 10, no. 1, pp. 188–196, 2022.
ISNAD Jafari Petroudi, Seyyed Hossein et al. “Some Special Matrices With Harmonic Numbers”. Konuralp Journal of Mathematics 10/1 (April 2022), 188-196.
JAMA Jafari Petroudi SH, Pirouz M, Akbıyık M, Yılmaz F. Some Special Matrices with Harmonic Numbers. Konuralp J. Math. 2022;10:188–196.
MLA Jafari Petroudi, Seyyed Hossein et al. “Some Special Matrices With Harmonic Numbers”. Konuralp Journal of Mathematics, vol. 10, no. 1, 2022, pp. 188-96.
Vancouver Jafari Petroudi SH, Pirouz M, Akbıyık M, Yılmaz F. Some Special Matrices with Harmonic Numbers. Konuralp J. Math. 2022;10(1):188-96.
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