Some Special Matrices with Harmonic Numbers

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.