On an infinite family of unipotent Sylvester–Kac-like matrices

Zhibin Du and Carlos M. da Fonseca
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 31, 2025, Number 4, Pages 846–850
DOI: 10.7546/nntdm.2025.31.4.846-850
Full paper (PDF, 156 Kb)

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Authors and affiliations

Zhibin Du
¹ School of Artificial Intelligence, South China Normal University
Foshan, Guangdong 528225, China

Carlos M. da Fonseca
² Kuwait College of Science and Technology
Doha District, Safat 13133, Kuwait
³ Faculty of Applied Mathematics and Informatics, Technical University of Sofia
Kliment Ohridski Blvd. 8, 1000 Sofia, Bulgaria
⁴ Chair of Computational Mathematics, University of Deusto
48007 Bilbao, Spain

Abstract

Classical Sylvester–Kac matrices are tridiagonal integral matrices with positive off-diagonal entries and fully integral spectra. Here, by relaxing the positivity requirement and using a lower Pascal triangle framework, we define, for each positive integer , a unipotent Sylvester–Kac-like matrix in which is the only eigenvalue. This construction highlights the connection to the original Sylvester–Kac matrices while introducing a new family of unipotent matrices with distinctive properties.

Keywords

• Sylvester–Kac matrix
• Tridiagonal matrix
• Unipotent matrix
• Eigenvalues

2020 Mathematics Subject Classification

• 15A15
• 15A18
• 15B36

References

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Manuscript history

• Received: 2 September 2025
• Revised: 14 October 2025
• Accepted: 10 November 2025
• Online First: 16 November 2025

Copyright information

Ⓒ 2025 by the Authors.
This is an Open Access paper distributed under the terms and conditions of the Creative Commons Attribution 4.0 International License (CC BY 4.0).