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

Baijuan Shi; Baijuan Shi

doi:10.3934/math.2022458

AIMS Mathematics

2022, Volume 7, Issue 5: 8224-8234. doi: 10.3934/math.2022458

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Research article

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

Baijuan Shi ^,

School of Science, Xi'an University of Posts and Telecommunications, 710121, ShaanXi, China

Received: 24 December 2021 Revised: 14 February 2022 Accepted: 17 February 2022 Published: 25 February 2022
MSC : 15A15, 15A60

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.
- particular matrix,
- exponential form,
- determinant,
- inversion,
- principal minor,
- norms
Citation: Baijuan Shi. A particular matrix with exponential form, its inversion and some norms[J]. AIMS Mathematics, 2022, 7(5): 8224-8234. doi: 10.3934/math.2022458

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Abstract

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.

References

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