A particular matrix with exponential form, its inversion and some norms

Abstract

<abstract><p>In this paper, we study a particular $ n\times n $ matrix $ A = [a_{k_{ij}}]^n_{i, j = 1} $ and its Hadamard inverse $ A^{\circ (-1)} $, whose entire elements are exponential form $ a_k = e(\frac{k}{n}) = e^{\frac{2\pi ik}{n}}, $ where $ k_{ij} = \min(i, j)+1 $. We study determinants, leading principal minor and inversions of $ A, $ $ A^{\circ (-1)} $. Then the defined values of Euclidean norms, $ l_p $ norms and spectral norms of these matrices are presented, rather than upper and lower bounds, which are different from other articles.</p></abstract>