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

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