Sylvester–Kac matrices with quadratic spectra: A comprehensive note

Sylvester–Kac matrices are tridiagonal integral matrices with integral spectra or with eigenvalues presenting some kind of regularity. Recently, several results have emerged independently in the literature with spectra having some type of quadratic form. In this note, we review those main results. U...

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Bibliographic Details
Published in:The Ramanujan journal Vol. 65; no. 3; pp. 1313 - 1322
Main Authors: Du, Zhibin, da Fonseca, Carlos M.
Format: Journal Article
Language:English
Published: New York Springer US 01.11.2024
Subjects:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematics
Mathematics and Statistics
Number Theory
Eigenvalues
15A15
15B36
Sylvester–Kac matrix
Tridiagonal matrix
15A18
ISSN:1382-4090, 1572-9303
Online Access:Get full text
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Summary:Sylvester–Kac matrices are tridiagonal integral matrices with integral spectra or with eigenvalues presenting some kind of regularity. Recently, several results have emerged independently in the literature with spectra having some type of quadratic form. In this note, we review those main results. Using a lower triangular matrix based on the Pascal’s triangle, we present an alternative unified approach to them. Ultimately, we provide a simple proof for Sylvester’s determinant claim. A comprehensive list of the major historical advances and generalizations, as well as the most recent contributions, is provided at the end.
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-024-00940-4