Open AccessA particular matrix with exponential form, its inversion and some norms
Author(s) -
Beibei Shi
Publication year - 2022
Publication title -
aims mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.329
H-Index - 15
ISSN - 2473-6988
DOI - 10.3934/math.2022458
Subject(s) - hadamard transform , mathematics , inverse , combinatorics , exponential function , matrix norm , matrix (chemical analysis) , hadamard matrix , minor (academic) , physics , eigenvalues and eigenvectors , mathematical analysis , quantum mechanics , geometry , law , chemistry , chromatography , political science
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.