Some Special Matrices with Harmonic Numbers

Amaç & Hipotez: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.

📊

Temel Bulgular & Veri:Determinants and inverses of these matrices are investigated.

💡

Sonuç & Katkı:Finally, we derive some identities about principal minors of these matrices.

🔬Tür:Araştırma Makalesi

📍Kapsam:Genişletilmiş Akademik Örneklem

⚙️Değişkenler:Harmonic number · Spectral norm · Hadamard inverse

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.

Anahtar kelimeler

Harmonic number Spectral norm Hadamard inverse determinant

🔖

Bu Makaleye Atıf Yap

PETROUDİ, S. H. J., PİROUZ, M., AKBIYIK, M., YILMAZ, F. (2022). Some Special Matrices with Harmonic Numbers. Konuralp Journal of Mathematics, Cilt: 10 - Sayı: 1, 188–196. https://ajindex.com/articles/786367

📄 Tam Metin PDF Okuyucu

Yeni Sekmede Aç ↗

↗Yeni Sekme

📄

PDF görüntüleyici hazırlanıyor...

Kaynakça (17)

M. Akbulak, A. Ipek, Hadamard exponential Hankel Matrix, its eigenvalues and some norms, Math. sci. Lett, 1 (2012), 81-87
R. Reams, Hadamard inverses, square roots and products of almost semidefinite matrices, Linear Algebra and its Applications, 288 (1999), 35-43
M. Bahsi, S. Solak, on the matrices with Harmonic numbers, GU. J. Sc, 23(4) (2010), 445-448
M. Bahsi, on the norms of r−circulant matrices wit the hyper Harmonic numbers, Journal of mathematical inequalities, 10(2) (2016), 445-458
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
D.Bozkurt, A note on the spectral norms of the matrices connected integer numbers sequence, Math.GM,. 1724v1 (2011), 171-190
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
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
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
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
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
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
S.H.J. Petroudi, B. Pirouz, A particular matrix, Its inversion and some norms, Appl. and Computational Math. 4(2) (2015), 47-52
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
S. Solak, Bahsi. M, A particular matrix and its some properties, Scientific Research and Essays, Vol.8(1), (2013), 1-5
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
N. Tuglu, C. Kizilates, S. Kesim, On the harmonic and hyperharmonic Fibonacci numbers, Advances in Difference Equations, (2015